Cosmology - damtp

CosmologyPart III Mathematical Tripos

13.8 billion yrs

380,000 yrs

10-34 sec

Daniel [email protected]

Contents

Preface 1

I The Homogeneous Universe 3

1 Geometry and Dynamics 4

1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Symmetric Three-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Distances∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 Matter Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.2 Spacetime Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.3 Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Inflation 29

2.1 The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.1 Light and Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.2 Growing Hubble Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.3 Why is the CMB so uniform? . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 A Shrinking Hubble Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 Solution of the Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.2 Hubble Radius vs. Particle Horizon . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Conditions for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 The Physics of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 Scalar Field Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.2 Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Reheating∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Thermal History 42

3.1 The Hot Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Local Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.2 Decoupling and Freeze-Out . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.3 A Brief History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Equilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.2 Densities and Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

i

Contents ii

3.2.3 Conservation of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.4 Neutrino Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.5 Electron-Positron Annihilation . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.6 Cosmic Neutrino Background . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Beyond Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.2 Dark Matter Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.3 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.4 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

II The Inhomogeneous Universe 76

4 Cosmological Perturbation Theory 77

4.1 Newtonian Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Perturbed Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.2 Jeans’ Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.3 Dark Matter inside Hubble . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Relativistic Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.1 Perturbed Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Perturbed Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.3 Linearised Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Conserved Curvature Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.1 Comoving Curvature Perturbation . . . . . . . . . . . . . . . . . . . . . . 96

4.3.2 A Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Structure Formation 101

5.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Superhorizon Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.1.2 Radiation-to-Matter Transition . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 Evolution of Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.1 Gravitational Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.3 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.4 Baryons∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Initial Conditions from Inflation 111

6.1 From Quantum to Classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Classical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Mukhanov-Sasaki Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.2 Subhorizon Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3 Quantum Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.1 Canonical Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.2 Choice of Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3.3 Zero-Point Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contents iii

6.4 Quantum Fluctuations in de Sitter Space . . . . . . . . . . . . . . . . . . . . . . 118

6.4.1 Canonical Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4.2 Choice of Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4.3 Zero-Point Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.4 Quantum-to-Classical Transition∗ . . . . . . . . . . . . . . . . . . . . . . . 121

6.5 Primordial Perturbations from Inflation . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.1 Curvature Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.6 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6.1 Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6.2 CMB Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Preface

This course is about 13.8 billion years of cosmic evolution:

At early times, the universe was hot and dense. Interactions between particles were frequent

and energetic. Matter was in the form of free electrons and atomic nuclei with light bouncing

between them. As the primordial plasma cooled, the light elements—hydrogen, helium and

lithium—formed. At some point, the energy had dropped enough for the first stable atoms

to exist. At that moment, photons started to stream freely. Today, billions of years later, we

observe this afterglow of the Big Bang as microwave radiation. This radiation is found to be

almost completely uniform, the same temperature (about 2.7 K) in all directions. Crucially, the

cosmic microwave background contains small variations in temperature at a level of 1 part in

10 000. Parts of the sky are slightly hotter, parts slightly colder. These fluctuations reflect tiny

variations in the primordial density of matter. Over time, and under the influence of gravity,

these matter fluctuations grew. Dense regions were getting denser. Eventually, galaxies, stars

and planets formed.

dark matterda

rk e

nerg

y

baryons

radiation

Inflation

Dark MatterProduction

Cosmic MicrowaveBackground

StructureFormation

Big BangNucleosynthesis

dark matter (27%)

dark energy (68%)

baryons (5%)

present energy density

fract

ion

of e

nerg

y de

nsity

1.0

0.0

(Chapter 3)

(Chapter 3)

(Chapters 2 and 6)(Chapter 3)

(Chapters 4 and 5)

0-10-20-30 10

010

13.8 Gyr380 kyr3 min

0.1 MeV 0.1 eV0.1 TeV

This picture of the universe—from fractions of a second after the Big Bang until today—

is a scientific fact. However, the story isn’t without surprises. The majority of the universe

today consists of forms of matter and energy that are unlike anything we have ever seen in

terrestrial experiments. Dark matter is required to explain the stability of galaxies and the rate

of formation of large-scale structures. Dark energy is required to rationalise the striking fact that

the expansion of the universe started to accelerate recently (meaning a few billion years ago).

What dark matter and dark energy are is still a mystery. Finally, there is growing evidence

that the primordial density perturbations originated from microscopic quantum fluctuations,

stretched to cosmic sizes during a period of inflationary expansion. The physical origin of

inflation is still a topic of active research.

1

2 Preface

Administrative comments.—Up-to-date versions of the lecture notes will be posted on the

course website:

www.damtp.cam.ac.uk/user/db275/cosmology.pdf

Starred sections (∗) are non-examinable.

Boxed text contains technical details and derivations that may be omitted on a first reading.

Please do not hesitate to email me questions, comments or corrections:

[email protected]

There will be four problem sets, which will appear in two-week intervals on the course website.

Details regarding supervisions will be announced in the lectures.

Notation and conventions.—We will mostly use natural units, in which the speed of light and

Planck’s constant are set equal to one, c = ~ ≡ 1. Length and time then have the same units.

Our metric signature is (+−−−), so that ds2 = dt2−dx2 for Minkowski space. This is the same

signature as used in the QFT course, but the opposite of the GR course. Spacetime four-vectors

will be denoted by capital letters, e.g. Xµ and Pµ, where the Greek indices µ, ν, · · · run from 0

to 3. We will use the Einstein summation convention where repeated indices are summed over.

Latin indices i, j, k, · · · will stand for spatial indices, e.g. xi and pi. Bold font will denote spatial

three-vectors, e.g. x and p.

Further reading.—I recommend the following textbooks:

. Dodelson, Modern Cosmology

A very readable book at about the same level as these lectures. My Boltzmann-centric treatment

of BBN and recombination was heavily inspired by Dodelson’s Chapter 3.

. Peter and Uzan, Primordial Cosmology

A recent book that contains a lot of useful reference material. Also good for Advanced Cosmology.

. Kolb and Turner, The Early Universe

A remarkably timeless book. It is still one of the best treatments of the thermal history of the

early universe.

. Weinberg, Cosmology

Written by the hero of a whole generation of theoretical physicists, this is the text to consult if you

are ever concerned about a lack of rigour. Unfortunately, Weinberg doesn’t do plots.

Acknowledgements.—Thanks to Paolo Creminelli for comments on a previous version of these

notes. Adam Solomon was a fantastic help in designing the problem sets and writing some of

the solutions.

Part I

The Homogeneous Universe

3

1 Geometry and Dynamics

The further out we look into the universe, the simpler it seems to get (see fig. 1.1). Averaged over

large scales, the clumpy distribution of galaxies becomes more and more isotropic—i.e. indepen-

dent of direction. Despite what your mom might have told you, we shouldn’t assume that we

are the centre of the universe. (This assumption is sometimes called the cosmological principle).

The universe should then appear isotropic to any (free-falling) observer throughout the universe.

If the universe is isotropic around all points, then it is also homogeneous—i.e. independent of

position. To a first approximation, we will therefore treat the universe as perfectly homogeneous

and isotropic. As we will see, in §1.1, homogeneity and isotropy single out a unique form of

the spacetime geometry. We discuss how particles and light propagate in this spacetime in §1.2.

Finally, in §1.3, we derive the Einstein equations and relate the rate of expansion of the universe

to its matter content.

Figure 1.1: The distribution of galaxies is clumpy on small scales, but becomes more uniform on large scales

and early times.

4

5 1. Geometry and Dynamics

1.1 Geometry

1.1.1 Metric

The spacetime metric plays a fundamental role in relativity. It turns observer-dependent coor-

dinates Xµ = (t, xi) into the invariant line element1

ds2 =3∑

µ,ν=0

gµνdXµdXν ≡ gµνdXµdXν . (1.1.1)

In special relativity, the Minkowski metric is the same everywhere in space and time,

gµν = diag(1,−1,−1,−1) . (1.1.2)

In general relativity, on the other hand, the metric will depend on where we are and when we

are,

gµν(t,x) . (1.1.3)

The spacetime dependence of the metric incorporates the effects of gravity. How the metric

depends on the position in spacetime is determined by the distribution of matter and energy in

the universe. For an arbitrary matter distribution, it can be next to impossible to find the metric

from the Einstein equations. Fortunately, the large degree of symmetry of the homogeneous

universe simplifies the problem.

flat

negatively curved

positively curved

Figure 1.2: The spacetime of the universe can be foliated into flat, positively curved or negatively curved

spatial hypersurfaces.

1.1.2 Symmetric Three-Spaces

Spatial homogeneity and isotropy mean that the universe can be represented by a time-ordered

sequence of three-dimensional spatial slices Σt, each of which is homogeneous and isotropic (see

fig. 1.2). We start with a classification of such maximally symmetric 3-spaces. First, we note that

homogeneous and isotropic 3-spaces have constant 3-curvature.2 There are only three options:

1Throughout the course, will use the Einstein summation convention where repeated indices are summed

over. We will also use natural units with c ≡ 1, so that dX0 = dt. Our metric signature will be mostly

minus, (+,−,−,−).2We give a precise definition of Riemann curvature below.


zero curvature, positive curvature and negative curvature. Let us determine the metric for each

case:

• flat space: the line element of three-dimensional Euclidean space E3 is simply

d`2 = dx2 = δijdxidxj . (1.1.4)

This is clearly invariant under spatial translations (xi 7→ xi + ai, with ai = const.) and

rotations (xi 7→ Rikxk, with δijR

ikR

jl = δkl).

• positively curved space: a 3-space with constant positive curvature can be represented as

a 3-sphere S3 embedded in four-dimensional Euclidean space E4,

d`2 = dx2 + du2 , x2 + u2 = a2 , (1.1.5)

where a is the radius of the 3-sphere. Homogeneity and isotropy of the surface of the

3-sphere are inherited from the symmetry of the line element under four-dimensional ro-

tations.

• negatively curved space: a 3-space with constant negative curvature can be represented as

a hyperboloid H3 embedded in four-dimensional Lorentzian space R1,3,

d`2 = dx2 − du2 , x2 − u2 = −a2 , (1.1.6)

where a2 is an arbitrary constant. Homogeneity and isotropy of the induced geometry

on the hyperboloid are inherited from the symmetry of the line element under four-

dimensional pseudo-rotations (i.e. Lorentz transformations, with u playing the role of time).

In the last two cases, it is convenient to rescale the coordinates, x→ ax and u→ au. The line

elements of the spherical and hyperbolic cases then are

d`2 = a2[dx2 ± du2

], x2 ± u2 = ±1 . (1.1.7)

Notice that the coordinates x and u are now dimensionless, while the parameter a carries

the dimension of length. The differential of the embedding condition, x2 ± u2 = ±1, gives

udu = ∓x · dx, so

d`2 = a2

[dx2 ± (x · dx)2

1∓ x2

]. (1.1.8)

We can unify (1.1.8) with the Euclidean line element (1.1.4) by writing

d`2 = a2

[dx2 + k

(x · dx)2

1− kx2

]≡ a2γij dxidxj , (1.1.9)

with

γij ≡ δij + kxixj

1− k(xkxk), for k ≡

0 Euclidean

+1 spherical

−1 hyperbolic

. (1.1.10)

Note that we must take a2 > 0 in order to have d`2 positive at x = 0, and hence everywhere.3

The form of the spatial metric γij depends on the choice of coordinates:

3Notice that despite appearance x = 0 is not a special point.


• It is convenient to use spherical polar coordinates, (r, θ, φ), because it makes the symme-

tries of the space manifest. Using

dx2 = dr2 + r2(dθ2 + sin2 θdφ2) , (1.1.11)

x · dx = rdr , (1.1.12)

the metric in (1.1.9) becomes diagonal

d`2 = a2

[dr2

1− kr2+ r2dΩ2

], (1.1.13)

where dΩ2 ≡ dθ2 + sin2 θdφ2.

• The complicated γrr component of (1.1.13) can sometimes be inconvenient. In that case,

we may redefine the radial coordinate, dχ ≡ dr/√

1− kr2, such that

d`2 = a2[dχ2 + S2

k(χ) dΩ2], (1.1.14)

where

Sk(χ) ≡

sinhχ k = −1

χ k = 0

sinχ k = +1

. (1.1.15)

1.1.3 Robertson-Walker Metric

To get the Robertson-Walker metric 4 for an expanding universe, we simply include d`2 =

a2γijdxidxj into the spacetime line element and let the parameter a be an arbitrary function of

time 5

ds2 = dt2 − a2(t)γijdxidxj . (1.1.16)

Notice that the symmetries of the universe have reduced the ten independent components of

the spacetime metric to a single function of time, the scale factor a(t), and a constant, the

curvature parameter k. The coordinates xi ≡ x1, x2, x3 are called comoving coordinates.

Fig. 1.3 illustrates the relation between comoving coordinates and physical coordinates, xiphys =

a(t)xi. The physical velocity of an object is

viphys ≡dxiphys

dt= a(t)

dxi

dt+da

dtxi ≡ vipec +Hxiphys . (1.1.17)

We see that this has two contributions: the so-called peculiar velocity, vipec ≡ a(t) xi, and the

Hubble flow, Hxiphys, where we have defined the Hubble parameter as 6

H ≡ a

a. (1.1.18)

The peculiar velocity of an object is the velocity measured by a comoving observer (i.e. an

observer who follows the Hubble flow).

4Sometimes this is called the Friedmann-Robertson-Walker (FRW) metric.5Skeptics might worry about uniqueness. Why didn’t we include a g0i component? Because it would break

isotropy. Why don’t we allow for a non-trivial g00 component? Because it can always be absorbed into a

redefinition of the time coordinate, dt′ ≡ √g00 dt.6Here, and in the following, an overdot denotes a time derivative, i.e. a ≡ da/dt.


time

Figure 1.3: Expansion of the universe. The comoving distance between points on an imaginary coordinate

grid remains constant as the universe expands. The physical distance is proportional to the comoving

distance times the scale factor a(t) and hence gets larger as time evolves.

• Using (1.1.13), the FRW metric in polar coordinates reads

ds2 = dt2 − a2(t)

[dr2

1− kr2+ r2dΩ2

]. (1.1.19)

This result is worth memorizing — after all, it is the metric of our universe! Notice that

the line element (1.1.19) has a rescaling symmetry

a→ λa , r → r/λ , k → λ2k . (1.1.20)

This means that the geometry of the spacetime stays the same if we simultaneously rescale

a, r and k as in (1.1.20). We can use this freedom to set the scale factor to unity today:7

a0 ≡ a(t0) ≡ 1. In this case, a(t) becomes dimensionless, and r and k−1/2 inherit the

dimension of length.

• Using (1.1.14), we can write the FRW metric as

ds2 = dt2 − a2(t)[dχ2 + S2

k(χ)dΩ2]. (1.1.21)

This form of the metric is particularly convenient for studying the propagation of light.

For the same purpose, it is also useful to introduce conformal time,

dτ =dt

a(t), (1.1.22)

so that (1.1.21) becomes

ds2 = a2(τ)[dτ2 −

(dχ2 + S2

k(χ)dΩ2)]. (1.1.23)

We see that the metric has factorized into a static Minkowski metric multiplied by a

time-dependent conformal factor a(τ). Since light travels along null geodesics, ds2 = 0,

the propagation of light in FRW is the same as in Minkowski space if we first transform

to conformal time. Along the path, the change in conformal time equals the change in

comoving distance,

∆τ = ∆χ . (1.1.24)

We will return to this in Chapter 2.

7Quantities that are evaluated at the present time t0 will have a subscript ‘0’.


1.2 Kinematics

1.2.1 Geodesics

How do particles evolve in the FRW spacetime? In the absence of additional non-gravitational

forces, freely-falling particles in a curved spacetime move along geodesics. I will briefly remind

you of some basic facts about geodesic motion in general relativity8 and then apply it to the

FRW spacetime (1.1.16).

Geodesic Equation∗

Consider a particle of mass m. In a curved spacetime it traces out a path Xµ(s). The four-

velocity of the particle is defined by

Uµ ≡ dXµ

ds. (1.2.25)

A geodesic is a curve which extremises the proper time ∆s/c between two points in spacetime.

In the box below, I show that this extremal path satisfies the geodesic equation 9

dUµ

ds+ ΓµαβU

αUβ = 0 , (1.2.26)

where Γµαβ are the Christoffel symbols,

Γµαβ ≡1

2gµλ(∂αgβλ + ∂βgαλ − ∂λgαβ) . (1.2.27)

Here, I have introduced the notation ∂µ ≡ ∂/∂Xµ. Moreover, you should recall that the inverse

metric is defined through gµλgλν = δµν .

Derivation of the geodesic equation.∗—Consider the motion of a massive particle between to points

in spacetime A and B (see fig. 1.4). The relativistic action of the particle is

S = −m∫ B

A

ds . (1.2.28)

Figure 1.4: Parameterisation of an arbitrary path in spacetime, Xµ(λ).

We label each point on the curve by a parameter λ that increases monotonically from an initial value

λ(A) ≡ 0 to a final value λ(B) ≡ 1. The action is a functional of the path Xµ(λ),

S[Xµ(λ)] = −m∫ 1

0

(gµν(X)XµXν

)1/2

dλ ≡∫ 1

0

L[Xµ, Xµ] dλ , (1.2.29)

8If all of this is new to you, you should arrange a crash-course with me and/or read Sean Carroll’s No-Nonsense

Introduction to General Relativity.9If you want to learn about the beautiful geometrical story behind geodesic motion I recommend Harvey Reall’s

Part III General Relativity lectures. Here, I simply ask you to accept the geodesic equation as the F = ma of

general relativity (for F = 0). From now on, we will use (1.2.26) as our starting point.


where Xµ ≡ dXµ/dλ. The motion of the particle corresponds to the extremum of this action. The

integrand in (1.2.29) is the Lagrangian L and it satisfies the Euler-Lagrange equation

d

dλ

(∂L

∂Xµ

)− ∂L

∂Xµ= 0 . (1.2.30)

The derivatives in (1.2.30) are

∂L

∂Xµ= − 1

LgµνX

ν ,∂L

∂Xµ= − 1

2L∂µgνρX

νXρ . (1.2.31)

Before continuing, it is convenient to switch from the general parameterisation λ to the parameteri-

sation using proper time s. (We could not have used s from the beginning since the value of s at B

is different for different curves. The range of integration would then have been different for different

curves.) Notice that (ds

dλ

)2

= gµνXµXν = L2 , (1.2.32)

and hence ds/dλ = L. In the above equations, we can therefore replace d/dλ with Ld/ds. The

Euler-Lagrange equation then becomes

d

ds

(gµν

dXν

ds

)− 1

2∂µgνρ

dXν

ds

dXρ

ds= 0 . (1.2.33)

Expanding the first term, we get

gµνd2Xν

ds2+ ∂ρgµν

dXρ

ds

dXν

ds− 1

2∂µgνρ

dXν

ds

dXρ

ds= 0 . (1.2.34)

In the second term, we can replace ∂ρgµν with 12 (∂ρgµν + ∂νgµρ) because it is contracted with an

object that is symmetric in ν and ρ. Contracting (1.2.34) with the inverse metric and relabelling

indices, we findd2Xµ

ds2+ Γµαβ

dXα

ds

dXβ

ds= 0 . (1.2.35)

Substituting (1.2.25) gives the desired result (1.2.26).

The derivative term in (1.2.26) can be manipulated by using the chain rule

d

dsUµ(Xα(s)) =

dXα

ds

∂Uµ

∂Xα= Uα

∂Uµ

∂Xα, (1.2.36)

so that we get

Uα(∂Uµ

∂Xα+ ΓµαβU

β

)= 0 . (1.2.37)

The term in brackets is the covariant derivative of Uµ, i.e. ∇αUµ ≡ ∂αUµ + ΓµαβUβ. This allows

us to write the geodesic equation in the following slick way: Uα∇αUµ = 0. In the GR course

you will derive this form of the geodesic equation directly by thinking about parallel transport.

Using the definition of the four-momentum of the particle,

Pµ = mUµ , (1.2.38)

we may also write (1.2.37) as

Pα∂Pµ

∂Xα= −ΓµαβP

αP β . (1.2.39)


For massless particles, the action (1.2.29) vanishes identically and our derivation of the geodesic

equation breaks down. We don’t have time to go through the more subtle derivation of the

geodesic equation for massless particles. Luckily, we don’t have to because the result is exactly

the same as (1.2.39).10 We only need to interpret Pµ as the four-momentum of a massless

particle.

Accepting that the geodesic equation (1.2.39) applies to both massive and massless particles,

we will move on. I will now show you how to apply the geodesic equation to particles in the

FRW universe.

Geodesic Motion in FRW

To evaluate the r.h.s. of (1.2.39) we need to compute the Christoffel symbols for the FRW

metric (1.1.16),

ds2 = dt2 − a2(t)γijdxidxj . (1.2.40)

All Christoffel symbols with two time indices vanish, i.e. Γµ00 = Γ00β = 0. The only non-zero

components are

Γ0ij = aaγij , Γi0j =

a

aδij , Γijk =

1

2γil(∂jγkl + ∂kγjl − ∂lγjk) , (1.2.41)

or are related to these by symmetry (note that Γµαβ = Γµβα). I will derive Γ0ij as an example and

leave Γi0j as an exercise.

Example.—The Christoffel symbol with upper index equal to zero is

Γ0αβ =

1

2g0λ(∂αgβλ + ∂βgαλ − ∂λgαβ) . (1.2.42)

The factor g0λ vanishes unless λ = 0 in which case it is equal to 1. Therefore,

Γ0αβ =

1

2(∂αgβ0 + ∂βgα0 − ∂0gαβ) . (1.2.43)

The first two terms reduce to derivatives of g00 (since gi0 = 0). The FRW metric has constant g00,

so these terms vanish and we are left with

Γ0αβ = −1

2∂0gαβ . (1.2.44)

The derivative is non-zero only if α and β are spatial indices, gij = −a2γij (don’t miss the sign!). In

that case, we find

Γ0ij = aaγij . (1.2.45)

The homogeneity of the FRW background implies ∂iPµ = 0, so that the geodesic equation (1.2.39)

reduces to

P 0dPµ

dt= −ΓµαβP

αP β ,

= −(

2Γµ0jP0 + ΓµijP

i)P j , (1.2.46)

10One way to think about massless particles is as the zero-mass limit of massive particles. A more rigorous

derivation of null geodesics from an action principle can be found in Paul Townsend’s Part III Black Holes lectures

[arXiv:gr-qc/9707012].


where I have used (1.2.41) in the second line.

• The first thing to notice from (1.2.46) is that massive particles at rest in the comoving

frame, P j = 0, will stay at rest because the r.h.s. then vanishes,

P j = 0 ⇒ dP i

dt= 0 . (1.2.47)

• Next, we consider the µ = 0 component of (1.2.46), but don’t require the particles to be

at rest. The first term on the r.h.s. vanishes because Γ00j = 0. Using (1.2.41), we then find

EdE

dt= −Γ0

ijPiP j = − a

ap2 , (1.2.48)

where we have written P 0 ≡ E and defined the amplitude of the physical three-momentum

as

p2 ≡ −gijP iP j = a2γijPiP j . (1.2.49)

Notice the appearance of the scale factor in (1.2.49) from the contraction with the spatial

part of the FRW metric, gij = −a2γij . The components of the four-momentum satisfy

the constraint gµνPµP ν = m2, or E2 − p2 = m2, where the r.h.s. vanishes for massless

particles. It follows that EdE = pdp, so that (1.2.48) can be written as

p

p= − a

a⇒ p ∝ 1

a. (1.2.50)

We see that the physical three-momentum of any particle (both massive and massless)

decays with the expansion of the universe.

– For massless particles, eq. (1.2.50) implies

p = E ∝ 1

a(massless particles) , (1.2.51)

i.e. the energy of massless particles decays with the expansion.

– For massive particles, eq. (1.2.50) implies

p =mv√1− v2

∝ 1

a(massive particles) , (1.2.52)

where vi = dxi/dt is the comoving peculiar velocity of the particles (i.e. the velocity

relative to the comoving frame) and v2 ≡ a2γijvivj is the magnitude of the physical

peculiar velocity, cf. eq. (1.1.17). To get the first equality in (1.2.52), I have used

P i = mU i = mdXi

ds= m

dt

dsvi =

mvi√1− a2γijvivj

=mvi√1− v2

. (1.2.53)

Eq. (1.2.52) shows that freely-falling particles left on their own will converge onto the

Hubble flow.


1.2.2 Redshift

Everything we know about the universe is inferred from the light we receive from distant ob-

jects. The light emitted by a distant galaxy can be viewed either quantum mechanically as

freely-propagating photons, or classically as propagating electromagnetic waves. To interpret

the observations correctly, we need to take into account that the wavelength of the light gets

stretched (or, equivalently, the photons lose energy) by the expansion of the universe. We now

quantify this effect.

Redshifting of photons.—In the quantum mechanical description, the wavelength of light is in-

versely proportional to the photon momentum, λ = h/p. Since according to (1.2.51) the mo-

mentum of a photon evolves as a(t)−1, the wavelength scales as a(t). Light emitted at time t1with wavelength λ1 will be observed at t0 with wavelength

λ0 =a(t0)

a(t1)λ1 . (1.2.54)

Since a(t0) > a(t1), the wavelength of the light increases, λ0 > λ1.

Redshifting of classical waves.—We can derive the same result by treating light as classical

electromagnetic waves. Consider a galaxy at a fixed comoving distance d. At a time τ1, the

galaxy emits a signal of short conformal duration ∆τ (see fig. 1.5). According to (1.1.24), the

light arrives at our telescopes at time τ0 = τ1 +d. The conformal duration of the signal measured

by the detector is the same as at the source, but the physical time intervals are different at the

points of emission and detection,

∆t1 = a(τ1)∆τ and ∆t0 = a(τ0)∆τ . (1.2.55)

If ∆t is the period of the light wave, the light is emitted with wavelength λ1 = ∆t1 (in units

where c = 1), but is observed with wavelength λ0 = ∆t0, so that

λ0

λ1=a(τ0)

a(τ1). (1.2.56)

Figure 1.5: In conformal time, the period of a light wave (∆τ) is equal at emission (τ1) and at observation (τ0).

However, measured in physical time (∆t = a(τ)∆τ) the period is longer when it reaches us, ∆t0 > ∆t1. We

say that the light has redshifted since its wavelength is now longer, λ0 > λ1.

It is conventional to define the redshift parameter as the fractional shift in wavelength of a

photon emitted by a distant galaxy at time t1 and observed on Earth today,

z ≡ λ0 − λ1

λ1. (1.2.57)


We then find

1 + z =a(t0)

a(t1). (1.2.58)

It is also common to define a(t0) ≡ 1, so that

1 + z =1

a(t1). (1.2.59)

Hubble’s law.—For nearby sources, we may expand a(t1) in a power series,

a(t1) = a(t0)[1 + (t1 − t0)H0 + · · ·

], (1.2.60)

where H0 is the Hubble constant

H0 ≡a(t0)

a(t0). (1.2.61)

Eq. (1.2.58) then gives z = H0(t0 − t1) + · · · . For close objects, t0 − t1 is simply the physical

distance d (in units with c = 1). We therefore find that the redshift increases linearly with

distance

z ' H0d . (1.2.62)

The slope in a redshift-distance diagram (cf. fig. 1.8) therefore measures the current expansion

rate of the universe, H0. These measurements used to come with very large uncertainties. Since

H0 normalizes everything else (see below), it became conventional to define11

H0 ≡ 100h kms−1Mpc−1 , (1.2.63)

where the parameter h is used to keep track of how uncertainties in H0 propagate into other

cosmological parameters. Today, measurements of H0 have become much more precise,12

h ≈ 0.67± 0.01 . (1.2.64)

1.2.3 Distances∗

For distant objects, we have to be more careful about what we mean by “distance”:

• Metric distance.—We first define a distance that isn’t really observable, but that will be

useful in defining observable distances. Consider the FRW metric in the form (1.1.21),

ds2 = dt2 − a2(t)[dχ2 + S2

k(χ)dΩ2], (1.2.65)

where13

Sk(χ) ≡

R0 sinh(χ/R0) k = −1

χ k = 0

R0 sin(χ/R0) k = +1

. (1.2.66)

The distance multiplying the solid angle element dΩ2 is the metric distance,

dm = Sk(χ) . (1.2.67)

11A parsec (pc) is 3.26 light-years. Blame astronomers for the funny units in (6.3.29).12Planck 2013 Results – Cosmological Parameters [arXiv:1303.5076].13Notice that the definition of Sk(χ) contains a length scale R0 after we chose to make the scale factor dimen-

sionless, a(t0) ≡ 1. This is achieved by using the rescaling symmetry a→ λa, χ→ χ/λ, and S2k → S2

k/λ.


In a flat universe (k = 0), the metric distance is simply equal to the comoving distance χ.

The comoving distance between us and a galaxy at redshift z can be written as

χ(z) =

∫ t0

t1

dt

a(t)=

∫ z

0

dz

H(z), (1.2.68)

where the redshift evolution of the Hubble parameter, H(z), depends on the matter content

of the universe (see §1.3). We emphasize that the comoving distance and the metric

distance are not observables.

• Luminosity distance.—Type IA supernovae are called ‘standard candles’ because they are

believed to be objects of known absolute luminosity L (= energy emitted per second).

The observed flux F (= energy per second per receiving area) from a supernova explosion

can then be used to infer its (luminosity) distance. Consider a source at a fixed comoving

distance χ. In a static Euclidean space, the relation between absolute luminosity and

observed flux is

F =L

4πχ2. (1.2.69)

source

observer

Figure 1.6: Geometry associated with the definition of luminosity distance.

In an FRW spacetime, this result is modified for three reasons:

1. At the time t0 that the light reaches the Earth, the proper area of a sphere drawn

around the supernova and passing through the Earth is 4πd2m. The fraction of the

light received in a telescope of aperture A is therefore A/4πd2m.

2. The rate of arrival of photons is lower than the rate at which they are emitted by the

redshift factor 1/(1 + z).

3. The energy E0 of the photons when they are received is less than the energy E1 with

which they were emitted by the same redshift factor 1/(1 + z).

Hence, the correct formula for the observed flux of a source with luminosity L at coordinate

distance χ and redshift z is

F =L

4πd2m(1 + z)2

≡ L

4πd2L

, (1.2.70)

where we have defined the luminosity distance, dL, so that the relation between luminosity,

flux and luminosity distance is the same as in (1.2.69). Hence, we find

dL = dm(1 + z) . (1.2.71)


• Angular diameter distance.—Sometimes we can make use of ‘standard rulers’, i.e. objects

of known physical size D. (This is the case, for example, for the fluctuations in the CMB.)

Let us assume again that the object is at a comoving distance χ and the photons which

we observe today were emitted at time t1. A naive astronomer could decide to measure

the distance dA to the object by measuring its angular size δθ and using the Euclidean

formula for its distance,14

dA =D

δθ. (1.2.72)

This quantity is called the angular diameter distance. The FRW metric (1.1.23) implies

source

observer

Figure 1.7: Geometry associated with the definition of angular diameter distance.

the following relation between the physical (transverse) size of the object and its angular

size on the sky

D = a(t1)Sk(χ)δθ =dm

1 + zδθ . (1.2.73)

Hence, we get

dA =dm

1 + z. (1.2.74)

The angular diameter distance measures the distance between us and the object when

the light was emitted. We see that angular diameter and luminosity distances aren’t

independent, but related by

dA =dL

(1 + z)2. (1.2.75)

Fig. 1.8 shows the redshift dependence of the three distance measures dm, dL, and dA. Notice

that all three distances are larger in a universe with dark energy (in the form of a cosmological

constant Λ) than in one without. This fact was employed in the discovery of dark energy (see

fig. 1.9 in §1.3.3).

1.3 Dynamics

The dynamics of the universe is determined by the Einstein equation

Gµν = 8πGTµν . (1.3.76)

This relates the Einstein tensor Gµν (a measure of the “spacetime curvature” of the FRW

universe) to the stress-energy tensor Tµν (a measure of the “matter content” of the universe). We

14This formula assumes δθ 1 (in radians) which is true for all cosmological objects.


with

without

distance

redshiftFigure 1.8: Distance measures in a flat universe, with matter only (dotted lines) and with 70% dark energy

(solid lines). In a dark energy dominated universe, distances out to a fixed redshift are larger than in a

matter-dominated universe.

will first discuss possible forms of cosmological stress-energy tensors Tµν (§1.3.1), then compute

the Einstein tensor Gµν for the FRW background (§1.3.2), and finally put them together to solve

for the evolution of the scale factor a(t) as a function of the matter content (§1.3.3).

1.3.1 Matter Sources

We first show that the requirements of isotropy and homogeneity force the coarse-grained stress-

energy tensor to be that of a perfect fluid,

Tµν = (ρ+ P )UµUν − P gµν , (1.3.77)

where ρ and P are the energy density and the pressure of the fluid and Uµ is its four-velocity

(relative to the observer).

Number Density

In fact, before we get to the stress-energy tensor, we study a simpler object: the number current

four-vector Nµ. The µ = 0 component, N0, measures the number density of particles, where for

us a “particle” may be an entire galaxy. The µ = i component, N i, is the flux of the particles in

the direction xi. Isotropy requires that the mean value of any 3-vector, such as N i, must vanish,

and homogeneity requires that the mean value of any 3-scalar15, such as N0, is a function only

of time. Hence, the current of galaxies, as measured by a comoving observer, has the following

components

N0 = n(t) , N i = 0 , (1.3.78)

where n(t) is the number of galaxies per proper volume as measured by a comoving observer.

A general observer (i.e. an observer in motion relative to the mean rest frame of the particles),

would measure the following number current four-vector

Nµ = nUµ , (1.3.79)

where Uµ ≡ dXµ/ds is the relative four-velocity between the particles and the observer. Of

course, we recover the previous result (1.3.78) for a comoving observer, Uµ = (1, 0, 0, 0). For

15A 3-scalar is a quantity that is invariant under purely spatial coordinate transformations.


Uµ = γ(1, vi), eq. (1.3.79) gives the correctly boosted results. For instance, you may recall that

the boosted number density is γn. (The number density increases because one of the dimensions

of the volume is Lorentz contracted.)

The number of particles has to be conserved. In Minkowski space, this implies that the

evolution of the number density satisfies the continuity equation

N0 = −∂iN i , (1.3.80)

or, in relativistic notation,

∂µNµ = 0 . (1.3.81)

Eq. (1.3.81) is generalised to curved spacetimes by replacing the partial derivative ∂µ with a

covariant derivative ∇µ,16

∇µNµ = 0 . (1.3.82)

Eq. (1.3.82) reduces to (1.3.81) in the local intertial frame.

Covariant derivative.—The covariant derivative is an important object in differential geometry and it

is of fundamental importance in general relativity. The geometrical meaning of ∇µ will be discussed

in detail in the GR course. In this course, we will have to be satisfied with treating it as an operator

that acts in a specific way on scalars, vectors and tensors:

• There is no difference between the covariant derivative and the partial derivative if it acts on

a scalar

∇µf = ∂µf . (1.3.83)

• Acting on a contravariant vector, V ν , the covariant derivative is a partial derivative plus a

correction that is linear in the vector:

∇µV ν = ∂µVν + ΓνµλV

λ . (1.3.84)

Look carefully at the index structure of the second term. A similar definition applies to the

covariant derivative of covariant vectors, ων ,

∇µων = ∂µων − Γλµνωλ . (1.3.85)

Notice the change of the sign of the second term and the placement of the dummy index.

• For tensors with many indices, you just repeat (1.3.84) and (1.3.85) for each index. For each

upper index you introduce a term with a single +Γ, and for each lower index a term with a

single −Γ:

∇σTµ1µ2···µkν1ν2···νl = ∂σT

µ1µ2···µkν1ν2···νl

+ Γµ1σλT

λµ2···µkν1ν2···νl + Γµ2

σλTµ1λ···µk

ν1ν2···νl + · · ·− Γλσν1T

µ1µ2···µkλν2···νl − Γλσν2T

µ1µ2···µkν1λ···νl − · · · . (1.3.86)

This is the general expression for the covariant derivative. Luckily, we will only be dealing

with relatively simple tensors, so this monsterous expression will usually reduce to something

managable.

16If this is the first time you have seen a covariant derivative, this will be a bit intimidating. Find me to talk

about your fears.


Explicitly, eq. (1.3.82) can be written

∇µNµ = ∂µNµ + ΓµµλN

λ = 0 . (1.3.87)

Using (1.3.78), this becomesdn

dt+ Γii0n = 0 , (1.3.88)

and substituting (1.2.41), we find

n

n= −3

a

a⇒ n(t) ∝ a−3 . (1.3.89)

As expected, the number density decreases in proportion to the increase of the proper volume.

Energy-Momentum Tensor

We will now use a similar logic to determine what form of the stress-energy tensor Tµν is

consistent with the requirements of homogeneity and isotropy. First, we decompose Tµν into a

3-scalar, T00, 3-vectors, Ti0 and T0j , and a 3-tensor, Tij . As before, isotropy requires the mean

values of 3-vectors to vanish, i.e. Ti0 = T0j = 0. Moreover, isotropy around a point x = 0

requires the mean value of any 3-tensor, such as Tij , at that point to be proportional to δij and

hence to gij , which equals −a2δij at x = 0,

Tij(x = 0) ∝ δij ∝ gij(x = 0) . (1.3.90)

Homogeneity requires the proportionality coefficient to be only a function of time. Since this is

a proportionality between two 3-tensors, Tij and gij , it must remain unaffected by an arbitrary

transformation of the spatial coordinates, including those transformations that preserve the form

of gij while taking the origin into any other point. Hence, homogeneity and isotropy require the

components of the stress-energy tensor everywhere to take the form

T00 = ρ(t) , πi ≡ Ti0 = 0 , Tij = −P (t)gij(t,x) . (1.3.91)

It looks even nicer with mixed upper and lower indices

Tµν = gµλTλν =

ρ 0 0 0

0 −P 0 0

0 0 −P 0

0 0 0 −P

. (1.3.92)

This is the stress-energy tensor of a perfect fluid as seen by a comoving observer. More generally,

the stress-energy tensor can be written in the following, explicitly covariant, form

Tµν = (ρ+ P )UµUν − P δµν , (1.3.93)

where Uµ ≡ dXµ/ds is the relative four-velocity between the fluid and the observer, while ρ and

P are the energy density and pressure in the rest-frame of the fluid. Of course, we recover the

previous result (1.3.92) for a comoving observer, Uµ = (1, 0, 0, 0).

How do the density and pressure evolve with time? In Minkowski space, energy and momen-

tum are conserved. The energy density therefore satisfies the continuity equation ρ = −∂iπi,i.e. the rate of change of the density equals the divergence of the energy flux. Similarly, the


evolution of the momentum density satisfies the Euler equation, πi = ∂iP . These conservation

laws can be combined into a four-component conservation equation for the stress-energy tensor

∂µTµν = 0 . (1.3.94)

In general relativity, this is promoted to the covariant conservation equation

∇µTµν = ∂µTµν + ΓµµλT

λν − ΓλµνT

µλ = 0 . (1.3.95)

Eq. (1.3.95) reduces to (1.3.94) in the local intertial frame. This corresponds to four separate

equations (one for each ν). The evolution of the energy density is determined by the ν = 0

equation

∂µTµ

0 + ΓµµλTλ

0 − Γλµ0Tµλ = 0 . (1.3.96)

Since T i0 vanishes by isotropy, this reduces to

dρ

dt+ Γµµ0ρ− Γλµ0T

µλ = 0 . (1.3.97)

From eq. (1.2.41) we see that Γλµ0 vanishes unless λ and µ are spatial indices equal to each other,

in which case it is a/a. The continuity equation (1.3.97) therefore reads

ρ+ 3a

a(ρ+ P ) = 0 . (1.3.98)

Exercise.—Show that (1.3.98) can be written as, dU = −PdV , where U = ρV and V ∝ a3.

Cosmic Inventory

The universe is filled with a mixture of different matter components. It is useful to classify the

different sources by their contribution to the pressure:

• Matter

We will use the term “matter” to refer to all forms of matter for which the pressure is

much smaller than the energy density, |P | ρ. As we will show in Chapter 3, this is the

case for a gas of non-relativistic particles (where the energy density is dominated by the

mass). Setting P = 0 in (1.3.98) gives

ρ ∝ a−3 . (1.3.99)

This dilution of the energy density simply reflects the expansion of the volume V ∝ a3.

– Dark matter. Most of the matter in the universe is in the form of invisible dark

matter. This is usually thought to be a new heavy particle species, but what it really

is, we don’t know.

– Baryons. Cosmologists refer to ordinary matter (nuclei and electrons) as baryons.17

17Of course, this is technically incorrect (electrons are leptons), but nuclei are so much heavier than electrons

that most of the mass is in the baryons. If this terminology upsets you, you should ask your astronomer friends

what they mean by “metals”.


• Radiation

We will use the term “radiation” to denote anything for which the pressure is about a

third of the energy density, P = 13ρ. This is the case for a gas of relativistic particles, for

which the energy density is dominated by the kinetic energy (i.e. the momentum is much

bigger than the mass). In this case, eq. (1.3.98) implies

ρ ∝ a−4 . (1.3.100)

The dilution now includes the redshifting of the energy, E ∝ a−1.

– Photons. The early universe was dominated by photons. Being massless, they are al-

ways relativistic. Today, we detect those photons in the form of the cosmic microwave

background.

– Neutrinos. For most of the history of the universe, neutrinos behaved like radiation.

Only recently have their small masses become relevant and they started to behave

like matter.

– Gravitons. The early universe may have produced a background of gravitons (i.e. grav-

itational waves, see §6.5.2). Experimental efforts are underway to detect them.

• Dark energy

We have recently learned that matter and radiation aren’t enough to describe the evolution

of the universe. Instead, the universe today seems to be dominated by a mysterious negative

pressure component, P = −ρ. This is unlike anything we have ever encountered in the

lab. In particular, from eq. (1.3.98), we find that the energy density is constant,

ρ ∝ a0 . (1.3.101)

Since the energy density doesn’t dilute, energy has to be created as the universe expands.18

– Vacuum energy. In quantum field theory, this effect is actually predicted! The

ground state energy of the vacuum corresponds to the following stress-energy tensor

T vacµν = ρvacgµν . (1.3.102)

Comparison with eq. (1.3.93), show that this indeed implies Pvac = −ρvac. Unfortu-

nately, the predicted size of ρvac is completely off,

ρvac

ρobs∼ 10120 . (1.3.103)

– Something else? The failure of quantum field theory to explain the size of the

observed dark energy has lead theorists to consider more exotic possibilities (such

as time-varying dark energy and modifications of general relativity). In my opinion,

none of these ideas works very well.

18In a gravitational system this doesn’t have to violate the conservation of energy. It is the conservation equation

(1.3.98) that counts.


Cosmological constant.—The left-hand side of the Einstein equation (1.3.76) isn’t uniquely defined.

We can add the term −Λgµν , for some constant Λ, without changing the conservation of the stress

tensor, ∇µTµν = 0 (recall, or check, that ∇µgµν = 0). In other words, we could have written the

Einstein equation as

Gµν − Λgµν = 8πGTµν . (1.3.104)

Einstein, in fact, did add such a term and called it the cosmological constant. However, it has become

modern practice to move this term to the r.h.s. and treat it as a contribution to the stress-energy

tensor of the form

T (Λ)µν =

Λ

8πGgµν ≡ ρΛ gµν . (1.3.105)

This is of the same form as the stress-energy tensor from vacuum energy, eq. (1.3.102).

Summary

Most cosmological fluids can be parameterised in terms of a constant equation of state: w = P/ρ.

This includes cold dark matter (w = 0), radiation (w = 1/3) and vacuum energy (w = −1). In

that case, the solutions to (1.3.98) scale as

ρ ∝ a−3(1+w) , (1.3.106)

and hence

ρ ∝

a−3 matter

a−4 radiation

a0 vacuum

. (1.3.107)

1.3.2 Spacetime Curvature

We want to relate these matter sources to the evolution of the scale factor in the FRW met-

ric (1.1.14). To do this we have to compute the Einstein tensor on the l.h.s. of the Einstein

equation (1.3.76),

Gµν = Rµν −1

2Rgµν . (1.3.108)

We will need the Ricci tensor

Rµν ≡ ∂λΓλµν − ∂νΓλµλ + ΓλλρΓρµν − ΓρµλΓλνρ , (1.3.109)

and the Ricci scalar

R = Rµµ = gµνRµν . (1.3.110)

Again, there is a lot of beautiful geometry behind these definitions. We will simply keep plugging-

and-playing: given the Christoffel symbols (1.2.41) nothing stops us from computing (1.3.109).

We don’t need to calculate Ri0 = R0i, because it is a 3-vector, and therefore must vanish

due to the isotropy of the Robertson-Walker metric. (Try it, if you don’t believe it!) The

non-vanishing components of the Ricci tensor are

R00 = −3a

a, (1.3.111)

Rij = −

[a

a+ 2

(a

a

)2

+ 2k

a2

]gij . (1.3.112)


Notice that we had to find Rij ∝ gij to be consistent with homogeneity and isotropy.

Derivation of R00.—Setting µ = ν = 0 in (1.3.109), we have

R00 = ∂λΓλ00 − ∂0Γλ0λ + ΓλλρΓρ00 − Γρ0λΓλ0ρ , (1.3.113)

Since Christoffels with two time-components vanish, this reduces to

R00 = −∂0Γi0i − Γi0jΓj0i . (1.3.114)

where in the second line we have used that Christoffels with two time-components vanish. Using

Γi0j = (a/a)δij , we find

R00 = − d

dt

(3a

a

)− 3

(a

a

)2

= −3a

a. (1.3.115)

Derivation of Rij.∗—Evaluating (1.3.112) is a bit tedious. A useful trick is to compute Rij(x =

0) ∝ δij ∝ gij(x = 0) using (1.1.9) and then transform the resulting relation between 3-tensors to

general x.

We first read off the spatial metric from (1.1.9),

γij = δij +kxixj

1− k(xkxk). (1.3.116)

The key point is to think ahead and anticipate that we will set x = 0 at the end. This allows

us to drop many terms. You may be tempted to use γij(x = 0) = δij straight away. However, the

Christoffel symbols contain a derivative of the metric and the Riemann tensor has another derivative,

so there will be terms in the final answer with two derivatives acting on the metric. These terms get

a contribution from the second term in (1.3.116). However, we can ignore the denominator in the

second term of γij and use

γij = δij + kxixj . (1.3.117)

The difference in the final answer vanishes at x = 0 [do you see why?]. The derivative of (1.3.117) is

∂lγij = k (δlixj + δljxi) . (1.3.118)

With this, we can evaluate

Γijk =1

2γil (∂jγkl + ∂kγjl − ∂lγjk) . (1.3.119)

The inverse metric is γij = δij − kxixj , but the second term won’t contribute when we set x = 0 in

the end [do you see why?], so we are free to use γij = δij . Using (1.3.118) in (1.3.119), we then get

Γijk = kxiδjk . (1.3.120)

This vanishes at x = 0, but its derivative does not

Γijk(x = 0) = 0 , ∂lΓijk(x = 0) = kδilδjk . (1.3.121)

We are finally ready to evaluate the Ricci tensor Rij at x = 0

Rij(x = 0) ≡ ∂λΓλij − ∂jΓλiλ︸︷︷︸(A)

+ ΓλλρΓρij − ΓρiλΓλjρ︸︷︷︸

(B)

. (1.3.122)


Let us first look at the two terms labelled (B). Dropping terms that are zero at x = 0, I find

(B) = Γll0Γ0ij − Γ0

ilΓlj0 − Γli0Γ0

jl

= 3a

aaaδij − aaδij

a

aδlj −

a

aδljaaδjl

= a2δij . (1.3.123)

The two terms labelled (A) in (1.3.122) can be evaluated by using (1.3.121),

(A) = ∂0Γ0ij + ∂lΓ

lij − ∂jΓlil

= ∂0(aa)δij + kδllδij − kδljδil=(aa+ a2 + 2k

)δij . (1.3.124)

Hence, I get

Rij(x = 0) = (A) + (B)

=[aa+ 2a2 + 2k

]δij

= −

[a

a+ 2

(a

a

)2

+ 2k

a2

]gij(x = 0) . (1.3.125)

As a relation between tensors this holds for general x, so we get the promised result (1.3.112). To

be absolutely clear, I will never ask you to reproduce a nasty computation like this.

The Ricci scalar is

R = −6

[a

a+

(a

a

)2

+k

a2

]. (1.3.126)

Exercise.—Verify eq. (1.3.126).

The non-zero components of the Einstein tensor Gµν ≡ gµλGλν are

G00 = 3

[(a

a

)2

+k

a2

], (1.3.127)

Gij =

[2a

a+

(a

a

)2

+k

a2

]δij . (1.3.128)

Exercise.—Verify eqs. (1.3.127) and (1.3.128).

1.3.3 Friedmann Equations

We combine eqs. (1.3.127) and (1.3.128) with stress-tensor (1.3.92), to get the Friedmann equa-

tions, (a

a

)2

=8πG

3ρ− k

a2, (1.3.129)

a

a= −4πG

3(ρ+ 3P ) . (1.3.130)


Here, ρ and P should be understood as the sum of all contributions to the energy density and

pressure in the universe. We write ρr for the contribution from radiation (with ργ for photons

and ρν for neutrinos), ρm for the contribution by matter (with ρc for cold dark matter and ρbfor baryons) and ρΛ for the vacuum energy contribution. The first Friedmann equation is often

written in terms of the Hubble parameter, H ≡ a/a,

H2 =8πG

3ρ− k

a2. (1.3.131)

Let us use subscripts ‘0’ to denote quantities evaluated today, at t = t0. A flat universe (k = 0)

corresponds to the following critical density today

ρcrit,0 =3H2

0

8πG= 1.9× 10−29 h2 grams cm−3

= 2.8× 1011 h2MMpc−3

= 1.1× 10−5 h2 protons cm−3 . (1.3.132)

We use the critical density to define dimensionless density parameters

ΩI,0 ≡ρI,0ρcrit,0

. (1.3.133)

The Friedmann equation (1.3.131) can then be written as

H2(a) = H20

[Ωr,0

(a0

a

)4+ Ωm,0

(a0

a

)3+ Ωk,0

(a0

a

)2+ ΩΛ,0

], (1.3.134)

where we have defined a “curvature” density parameter, Ωk,0 ≡ −k/(a0H0)2. It should be noted

that in the literature, the subscript ‘0’ is normally dropped, so that e.g. Ωm usually denotes

the matter density today in terms of the critical density today. From now on we will follow

this convention and drop the ‘0’ subscripts on the density parameters. We will also use the

conventional normalization for the scale factor, a0 ≡ 1. Eq. (1.3.134) then becomes

H2

H20

= Ωra−4 + Ωma

−3 + Ωka−2 + ΩΛ . (1.3.135)

ΛCDM

Observations (see figs. 1.9 and 1.10) show that the universe is filled with radiation (‘r’), matter

(‘m’) and dark energy (‘Λ’):

|Ωk| ≤ 0.01 , Ωr = 9.4× 10−5 , Ωm = 0.32 , ΩΛ = 0.68 .

The equation of state of dark energy seems to be that of a cosmological constant, wΛ ≈ −1. The

matter splits into 5% ordinary matter (baryons, ‘b’) and 27% (cold) dark matter (CDM, ‘c’):

Ωb = 0.05 , Ωc = 0.27 .

We see that even today curvature makes up less than 1% of the cosmic energy budget. At earlier

times, the effects of curvature are then completely negligible (recall that matter and radiation

scale as a−3 and a−4, respectively, while the curvature contribution only increases as a−2). For

the rest of these lectures, I will therefore set Ωk ≡ 0. In Chapter 2, we will show that inflation

indeed predicts that the effects of curvature should be minuscule in the early universe (see also

Problem Set 2).


Low-z

SDSS

SNLSHST

0.2 0.4 0.6 0.8 1.0 1.40.0 1.2redshift

14

16

18

20

22

24

26

dist

ance

(app

aren

t mag

nitu

de) 0.680.32

0.001.00

Figure 1.9: Type IA supernovae and the discovery dark energy. If we assume a flat universe, then the

supernovae clearly appear fainter (or more distant) than predicted in a matter-only universe (Ωm = 1.0).

(SDSS = Sloan Digital Sky Survey; SNLS = SuperNova Legacy Survey; HST = Hubble Space Telescope.)

0.24 0.32 0.40 0.48

0.56

0.64

0.72

0.80+lensing+lensing+BAO

40

45

50

55

60

65

70

75

Figure 1.10: A combination CMB and LSS observations indicate that the spatial geometry of the universe

is flat. The energy density of the universe is dominated by a cosmological constant. Notice that the CMB

data alone cannot exclude a matter-only universe with large spatial curvature. The evidence for dark energy

requires additional input.

Single-Component Universe

The different scalings of radiation (a−4), matter (a−3) and vacuum energy (a0) imply that for

most of its history the universe was dominated by a single component (first radiation, then

matter, then vacuum energy; see fig. 1.11). Parameterising this component by its equation of

state wI captures all cases of interest. For a flat, single-component universe, the Friedmann

equation (1.3.135) reduces toa

a= H0

√ΩI a

− 32

(1+wI) . (1.3.136)


matter

radiation

cosmological constant

Figure 1.11: Evolution of the energy densities in the universe.

Integrating this equation, we obtain the time dependence of the scale factor

a(t) ∝

t2/3(1+wI) wI 6= −1

t2/3 MD

t1/2 RD

eHt wI = −1 ΛD

(1.3.137)

or, in conformal time,

a(τ) ∝

τ2/(1+3wI) wI 6= −1

τ2 MD

τ RD

(−τ)−1 wI = −1 ΛD

(1.3.138)

Exercise.—Derive eq. (1.3.138) from eq. (1.3.137).

Table 1.1 summarises the solutions for a flat universe during radiation domination (RD), matter

domination (MD) and dark energy domination (ΛD).

w ρ(a) a(t) a(τ)

RD 13 a−4 t1/2 τ

MD 0 a−3 t2/3 τ2

ΛD −1 a0 eHt −τ−1

Table 1.1: FRW solutions for a flat single-component universe.


Two-Component Universe∗

Matter and radiation were equally important at aeq ≡ Ωr/Ωm ≈ 3 × 10−4, which was shortly

before the cosmic microwave background was released (in §3.3.3, we will show that this happened

at arec ≈ 9× 10−4). It will be useful to have an exact solution describing the transition era. Let

us therefore consider a flat universe filled with a mixture of matter and radiation. To solve for

the evolution of the scale factor, it proves convenient to move to conformal time. The Friedmann

equations (1.3.129) and (1.3.130) then are

(a′)2 =8πG

3ρa4 , (1.3.139)

a′′ =4πG

3(ρ− 3P )a3 , (1.3.140)

where primes denote derivatives with respect to conformal time and

ρ ≡ ρm + ρr =ρeq

2

[(aeq

a

)3+(aeq

a

)4]. (1.3.141)

Exercise.—Derive eqs. (1.3.139) and (1.3.140). You will first need to convince yourself that a = a′/a

and a = a′′/a2 − (a′)2/a3.

Notice that radiation doesn’t contribute as a source term in eq. (1.3.140), ρr−3Pr = 0. Moreover,

since ρma3 = const. = 1

2ρeqa3eq, we can write eq. (1.3.140) as

a′′ =2πG

3ρeqa

3eq . (1.3.142)

This equation has the following solution

a(τ) =πG

3ρeqa

3eqτ

2 + Cτ +D . (1.3.143)

Imposing a(τ = 0) ≡ 0, fixes one integration constant, D = 0. We find the second integration

constant by substituting (1.3.143) and (1.3.141) into (1.3.139),

C =

(4πG

3ρeqa

4eq

)1/2

. (1.3.144)

Eq. (1.3.143) can then be written as

a(τ) = aeq

[(τ

τ?

)2

+ 2

(τ

τ?

)], (1.3.145)

where

τ? ≡(πG

3ρeqa

2eq

)−1/2

=τeq√2− 1

. (1.3.146)

For τ τeq, we recover the radiation-dominated limit, a ∝ τ , while for τ τeq, we agree with

the matter-dominated limit, a ∝ τ2.

2 Inflation

The FRW cosmology described in the previous chapter is incomplete. It doesn’t explain why the

universe is homogeneous and isotropic on large scales. In fact, the standard cosmology predicts

that the early universe was made of many causally disconnected regions of space. The fact that

these apparently disjoint patches of space have very nearly the same densities and temperatures

is called the horizon problem. In this chapter, I will explain how inflation—an early period of

accelerated expansion—drives the primordial universe towards homogeneity and isotropy, even

if it starts in a more generic initial state.

Throughout this chapter, we will trade Newton’s constant for the (reduced) Planck mass,

Mpl ≡√

~c8πG

= 2.4× 1018 GeV ,

so that the Friedmann equation (1.3.131) is written as H2 = ρ/(3M2pl).

2.1 The Horizon Problem

2.1.1 Light and Horizons

The size of a causal patch of space it determined by how far light can travel in a certain amount

of time. As we mentioned in §1.1.3, in an expanding spacetime the propagation of light (photons)

is best studied using conformal time. Since the spacetime is isotropic, we can always define the

coordinate system so that the light travels purely in the radial direction (i.e. θ = φ = const.).

The evolution is then determined by a two-dimensional line element1

ds2 = a2(τ)[dτ2 − dχ2

]. (2.1.1)

Since photons travel along null geodesics, ds2 = 0, their path is defined by

∆χ(τ) = ±∆τ , (2.1.2)

where the plus sign corresponds to outgoing photons and the minus sign to incoming photons.

This shows the main benefit of working with conformal time: light rays correspond to straight

lines at 45 angles in the χ-τ coordinates. If instead we had used physical time t, then the

light cones for curved spacetimes would be curved. With these preliminaries, we now define two

different types of cosmological horizons. One which limits the distances at which past events

can be observed and one which limits the distances at which it will ever be possible to observe

future events.

1For the radial coordinate χ we have used the parameterisation of (1.1.23), so that (2.1.1) is conformal to

two-dimensional Minkowski space and the curvature k of the three-dimensional spatial slices is absorbed into the

definition of the coordinate χ. Had we used the regular polar coordinate r, the two-dimensional line element

would have retained a dependence on k. For flat slices, χ and r are of course the same.

29

30 2. Inflation

particle horizon at p

p

event horizon at p

comoving particle outside the particle horizon at p

Figure 2.1: Spacetime diagram illustrating the concept of horizons. Dotted lines show the worldlines of

comoving objects. The event horizon is the maximal distance to which we can send signal. The particle

horizon is the maximal distance from which we can receive signals.

• Particle horizon.—Eq. (2.1.2) tells us that the maximal comoving distance that light can

travel between two times τ1 and τ2 > τ1 is simply ∆τ = τ2−τ1 (recall that c ≡ 1). Hence, if

the Big Bang ‘started’ with the singularity at ti ≡ 0,2 then the greatest comoving distance

from which an observer at time t will be able to receive signals travelling at the speed of

light is given by

χph(τ) = τ − τi =

∫ t

ti

dt

a(t). (2.1.3)

This is called the (comoving) particle horizon. The size of the particle horizon at time τ

may be visualised by the intersection of the past light cone of an observer p with the

spacelike surface τ = τi (see fig. 2.1). Causal influences have to come from within this

region. Only comoving particles whose worldlines intersect the past light cone of p can

send a signal to an observer at p. The boundary of the region containing such worldlines

is the particle horizon at p. Notice that every observer has his of her own particle horizon.

• Event horizon.—Just as there are past events that we cannot see now, there may be future

events that we will never be able to see (and distant regions that we will never be able to

influence). In comoving coordinates, the greatest distance from which an observer at time

tf will receive signals emitted at any time later than t is given by

χeh(τ) = τf − τ =

∫ tf

t

dt

a(t). (2.1.4)

This is called the (comoving) event horizon. It is similar to the event horizon of black

holes. Here, τf denotes the ‘final moment of (conformal) time’. Notice that this may be

finite even if physical time is infinite, tf = +∞. Whether this is the case or not depends

on the form of a(t). In particular, τf is finite for our universe, if dark energy is really a

cosmological constant.

2Notice that the Big Bang singularity is a moment in time, but not a point space. Indeed, in figs. 2.1 and 2.2

we describe the singularity by an extended (possibly infinite) spacelike hypersurface.

31 2. Inflation

2.1.2 Growing Hubble Sphere

It is the particle horizon that is relevant for the horizon problem of the standard Big Bang

cosmology. Eq. (2.1.3) can be written in the following illuminating way

χph(τ) =

∫ t

ti

dt

a=

∫ a

ai

da

aa=

∫ ln a

ln ai

(aH)−1 d ln a , (2.1.5)

where ai ≡ 0 corresponds to the Big Bang singularity. The causal structure of the spacetime

can hence be related to the evolution of the comoving Hubble radius (aH)−1. For a universe

dominated by a fluid with constant equation of state w ≡ P/ρ, we get

(aH)−1 = H−10 a

12

(1+3w) . (2.1.6)

Note the dependence of the exponent on the combination (1 + 3w). All familiar matter sources

satisfy the strong energy condition (SEC), 1 + 3w > 0, so it used to be a standard assumption

that the comoving Hubble radius increases as the universe expands. In this case, the integral in

(2.1.5) is dominated by the upper limit and receives vanishing contributions from early times.

We see this explicitly in the example of a perfect fluid. Using (2.1.6) in (2.1.5), we find

χph(a) =2H−1

0

(1 + 3w)

[a

12

(1+3w) − a12

(1+3w)

i

]≡ τ − τi . (2.1.7)

The fact that the comoving horizon receives its largest contribution from late times can be made

manifest by defining

τi ≡2H−1

0

(1 + 3w)a

12

(1+3w)

i

ai→0 , w>− 13−−−−−−−−−−−→ 0 . (2.1.8)

The comoving horizon is finite,

χph(t) =2H−1

0

(1 + 3w)a(t)

12

(1+3w) =2

(1 + 3w)(aH)−1 . (2.1.9)

We see that in the standard cosmology χph ∼ (aH)−1. This has lead to the confusing practice

of referring to both the particle horizon and the Hubble radius as the “horizon” (see §2.2.2).

2.1.3 Why is the CMB so uniform?

About 380 000 years after the Big Bang, the universe had cooled enough to allow the formation

of hydrogen atoms and the decoupling of photons from the primordial plasma (see §3.3.3). We

observe this event in the form of the cosmic microwave background (CMB), an afterglow of the

hot Big Bang. Remarkably, this radiation is almost perfectly isotropic, with anisotropies in the

CMB temperature being smaller than one part in ten thousand.

A moment’s thought will convince you that the finiteness of the conformal time elapsed

between ti = 0 and the time of the formation of the CMB, trec, implies a serious problem: it

means that most spots in the CMB have non-overlapping past light cones and hence never were

in causal contact. This is illustrated by the spacetime diagram in fig. 2.2. Consider two opposite

directions on the sky. The CMB photons that we receive from these directions were emitted at

the points labelled p and q in fig. 2.2. We see that the photons were emitted sufficiently close to

the Big Bang singularity that the past light cones of p and q don’t overlap. This implies that

no point lies inside the particle horizons of both p and q. So the puzzle is: how do the photons

32 2. Inflation

coming from p and q “know” that they should be at almost exactly the same temperature? The

same question applies to any two points in the CMB that are separated by more than 1 degree

in the sky. The homogeneity of the CMB spans scales that are much larger than the particle

horizon at the time when the CMB was formed. In fact, in the standard cosmology the CMB is

made of about 104 disconnected patches of space. If there wasn’t enough time for these regions

to communicate, why do they look so similar? This is the horizon problem.

1100 10 3 1 0 1 1100

0.2

0.4

0.60.81.0

0.01

0.1

0.001

Hubb

le sp

here

now

light c

one

comoving distance [Glyr]

scal

e fa

ctor

conf

orm

al ti

me

[Gyr

]

-40 -20 0 20 40

50

40

30

20

10

0

3 10

CMB

our worldline

p q

Figure 2.2: The horizon problem in the conventional Big Bang model. All events that we currently observe

are on our past light cone. The intersection of our past light cone with the spacelike slice labelled CMB

corresponds to two opposite points in the observed CMB. Their past light cones don’t overlap before they

hit the singularity, a = 0, so the points appear never to have been in causal contact. The same applies to

any two points in the CMB that are separated by more than 1 degree on the sky.

2.2 A Shrinking Hubble Sphere

Our description of the horizon problem has highlighted the fundamental role played by the

growing Hubble sphere of the standard Big Bang cosmology. A simple solution to the horizon

problem therefore suggests itself: let us conjecture a phase of decreasing Hubble radius in the

early universe,d

dt(aH)−1 < 0 . (2.2.10)

If this lasts long enough, the horizon problem can be avoided. Physically, the shrinking Hubble

sphere requires a SEC-violating fluid, 1 + 3w < 0.

2.2.1 Solution of the Horizon Problem

For a shrinking Hubble sphere, the integral in (2.1.5) is dominated by the lower limit. The Big

Bang singularity is now pushed to negative conformal time,

τi =2H−1

0

(1 + 3w)a

12

(1+3w)

i

ai→0 , w<− 13−−−−−−−−−−−→ −∞ . (2.2.11)

This implies that there was “much more conformal time between the singularity and decoupling

than we had thought”! Fig. 2.3 shows the new spacetime diagram. The past light cones of

33 2. Inflation

1100 10 3 1 0 1 1100

0.2

0.4

0.60.81.0

0.01

0.1

0.001

Hubb

le sp

here

now

light c

one

scal

e fa

ctor

conf

orm

al ti

me

[Gyr

]

50

40

30

20

10

3 10

CMBreheating

-10

-20

-30

-40

infla

tion

causal contact

Figure 2.3: Inflationary solution to the horizon problem. The comoving Hubble sphere shrinks during

inflation and expands during the conventional Big Bang evolution (at least until dark energy takes over at

a ≈ 0.5). Conformal time during inflation is negative. The spacelike singularity of the standard Big Bang is

replaced by the reheating surface, i.e. rather than marking the beginning of time it now corresponds simply

to the transition from inflation to the standard Big Bang evolution. All points in the CMB have overlapping

past light cones and therefore originated from a causally connected region of space.

widely separated points in the CMB now had enough time to intersect before the time τi. The

uniformity of the CMB is not a mystery anymore. In inflationary cosmology, τ = 0 isn’t the

initial singularity, but instead becomes only a transition point between inflation and the standard

Big Bang evolution. There is time both before and after τ = 0.

2.2.2 Hubble Radius vs. Particle Horizon

A quick word of warning about bad (but unfortunately standard) language in the inflationary

literature: Both the particle horizon χph and the Hubble radius (aH)−1 are often referred to

simply as the “horizon”. In the standard FRW evolution (with ordinary matter) the two are

roughly the same—cf. eq. (2.1.9)—so giving them the same name isn’t an issue. However, the

whole point of inflation is to make the particle horizon much larger than the Hubble radius.

The Hubble radius (aH)−1 is the (comoving) distance over which particles can travel in the

course of one expansion time.3 It is therefore another way of measuring whether particles are

causally connected with each other: comparing the comoving separation λ of two particles with

(aH)−1 determines whether the particles can communicate with each other at a given moment

(i.e. within the next Hubble time). This makes it clear that χph and (aH)−1 are conceptually

very different:

3The expansion time, tH ≡ H−1 = dt/d ln a, is roughly the time in which the scale factor doubles.

34 2. Inflation

• if λ > χph, then the particles could never have communicated.

• if λ > (aH)−1, then the particles cannot talk to each other now.

Inflation is a mechanism to achieve χph (aH)−1. This means that particles can’t communi-

cate now (or when the CMB was created), but were in causal contact early on. In particular,

the shrinking Hubble sphere means that particles which were initially in causal contact with

another—i.e. separated by a distance λ < (aIHI)−1—can no longer communicate after a suf-

ficiently long period of inflation: λ > (aH)−1; see fig. 2.4. However, at any moment before

horizon exit (careful: I really mean exit of the Hubble radius!) the particles could still talk

to each other and establish similar conditions. Everything within the Hubble sphere at the

beginning of inflation, (aIHI)−1, was causally connected.

Since the Hubble radius is easier to calculate than the particle horizon it is common to use

the Hubble radius as a means of judging the horizon problem. If the entire observable universe

was within the comoving Hubble radius at the beginning of inflation—i.e. (aIHI)−1 was larger

than the comoving radius of the observable universe (a0H0)−1—then there is no horizon problem.

Notice that this is more conservative than using the particle horizon since χph(t) is always bigger

than (aH)−1(t). Moreover, using (aIHI)−1 as a measure of the horizon problem means that we

don’t have to assume anything about earlier times t < tI .

time

scales

reheatinginflation “ Big Bang ”

standard Big Bang

inflation

Figure 2.4: Scales of cosmological interest were larger than the Hubble radius until a ≈ 10−5 (where today is

at a(t0) ≡ 1). However, at very early times, before inflation operated, all scales of interest were smaller than

the Hubble radius and therefore susceptible to microphysical processing. Similarly, at very late times, the

scales of cosmological interest are back within the Hubble radius. Notice the symmetry of the inflationary

solution. Scales just entering the horizon today, 60 e-folds after the end of inflation, left the horizon 60

e-folds before the end of inflation.

Duration of inflation.—How much inflation do we need to solve the horizon problem? At the very

least, we require that the observable universe today fits in the comoving Hubble radius at the begin-

ning of inflation,

(a0H0)−1 < (aIHI)−1 . (2.2.12)

Let us assume that the universe was radiation dominated since the end of inflation and ignore the

relatively recent matter- and dark energy-dominated epochs. Remembering that H ∝ a−2 during

radiation domination, we have

a0H0

aEHE∼ a0

aE

(aEa0

)2

=aEa0∼ T0

TE∼ 10−28 , (2.2.13)

35 2. Inflation

where in the numerical estimate we used TE ∼ 1015 GeV and T0 = 10−3 eV (∼ 2.7 K). Eq. (2.2.12)

can then be written as

(aIHI)−1 > (a0H0)−1 ∼ 1028(aEHE)−1 . (2.2.14)

For inflation to solve the horizon problem, (aH)−1 should therefore shrink by a factor of 1028. The

most common way to arrange this it to have H ∼ const. during inflation (see below). This implies

HI ≈ HE , so eq. (2.2.14) becomes

aEaI

> 1028 ⇒ ln

(aEaI

)> 64 . (2.2.15)

This is the famous statement that the solution of the horizon problem requires about 60 e-folds of

inflation.

2.2.3 Conditions for Inflation

I like the shrinking Hubble sphere as the fundamental definition of inflation since it relates most

directly to the horizon problem and is also key for the inflationary mechanism of generating

fluctuations (see Chapter 6). However, before we move on to discuss what physics can lead to a

shrinking Hubble sphere, let me show you that this definition of inflation is equivalent to other

popular ways of describing inflation.

• Accelerated expansion.—From the relation

d

dt(aH)−1 =

d

dt(a)−1 = − a

(a)2, (2.2.16)

we see that a shrinking comoving Hubble radius implies accelerated expansion

a > 0 . (2.2.17)

This explains why inflation is often defined as a period of acceleration.

• Slowly-varying Hubble parameter.—Alternatively, we may write

d

dt(aH)−1 = − aH + aH

(aH)2= −1

a(1− ε) , where ε ≡ − H

H2. (2.2.18)

The shrinking Hubble sphere therefore also corresponds to

ε = − H

H2< 1 . (2.2.19)

• Quasi-de Sitter expansion.—For perfect inflation, ε = 0, the spacetime becomes de Sitter

space

ds2 = dt2 − e2Htdx2 , (2.2.20)

where H = ∂t ln a = const. Inflation has to end, so it shouldn’t correspond to perfect de

Sitter space. However, for small, but finite ε 6= 0, the line element (2.2.20) is still a good

approximation to the inflationary background. This is why we will often refer to inflation

as a quasi-de Sitter period.

36 2. Inflation

• Negative pressure.—What forms of stress-energy source accelerated expansion? Let us

consider a perfect fluid with pressure P and density ρ. The Friedmann equation, H2 =

ρ/(3M2pl), and the continuity equation, ρ = −3H(ρ+ P ), together imply

H +H2 = − 1

6M2pl

(ρ+ 3P ) = −H2

2

(1 +

3P

ρ

). (2.2.21)

We rearrange this to find that

ε = − H

H2=

3

2

(1 +

P

ρ

)< 1 ⇔ w ≡ P

ρ< −1

3, (2.2.22)

i.e. inflation requires negative pressure or a violation of the strong energy condition. How

this can arise in a physical theory will be explained in the next section. We will see that

there is nothing sacred about the strong energy condition and that it can easily be violated.

• Constant density.—Combining the continuitiy equation, ρ = −3H(ρ+P ), with eq. (2.2.21),

we find ∣∣∣∣d ln ρ

d ln a

∣∣∣∣ = 2ε < 1 . (2.2.23)

For small ε, the energy density is therefore nearly constant. Conventional matter sources

all dilute with expansion, so we need to look for something more unusual.

2.3 The Physics of Inflation

We have shown that a given FRW spacetime with time-dependent Hubble parameter H(t)

corresponds to cosmic acceleration if and only if

ε ≡ − H

H2= −d lnH

dN< 1 . (2.3.24)

Here, we have defined dN ≡ d ln a = Hdt, which measures the number of e-foldsN of inflationary

expansion. Eq. (2.3.24) implies that the fractional change of the Hubble parameter per e-fold is

small. Moreover, in order to solve the horizon problem, we want inflation to last for a sufficiently

long time (usually at least N ∼ 40 to 60 e-folds). To achieve this requires ε to remain small for

a sufficiently large number of Hubble times. This condition is measured by a second parameter

η ≡ d ln ε

dN=

ε

Hε. (2.3.25)

For |η| < 1, the fractional change of ε per Hubble time is small and inflation persists. In this

section, we discuss what microscopic physics can lead to the conditions ε < 1 and |η| < 1.

2.3.1 Scalar Field Dynamics

As a simple toy model for inflation we consider a scalar field, the inflaton φ(t,x). As indicated by

the notation, the value of the field can depend on time t and the position in space x. Associated

with each field value is a potential energy density V (φ) (see fig. 2.5). If the field is dynamical

(i.e. changes with time) then it also carries kinetic energy density. If the stress-energy associated

with the scalar field dominates the universe, it sources the evolution of the FRW background.

We want to determine under which conditions this can lead to accelerated expansion.

37 2. Inflation

Figure 2.5: Example of a slow-roll potential. Inflation occurs in the shaded parts of the potential.

The stress-energy tensor of the scalar field is 4

Tµν = ∂µφ∂νφ− gµν(

1

2gαβ∂αφ∂βφ− V (φ)

). (2.3.26)

Consistency with the symmetries of the FRW spacetime requires that the background value of

the inflaton only depends on time, φ = φ(t). From the time-time component T 00 = ρφ, we infer

that

ρφ =1

2φ2 + V (φ) . (2.3.27)

We see that the total energy density, ρφ, is simply the sum of the kinetic energy density, 12 φ

2,

and the potential energy density, V (φ). From the space-space component T ij = −Pφ δij , we find

that the pressure is the difference of kinetic and potential energy densities,

Pφ =1

2φ2 − V (φ) . (2.3.28)

We see that a field configuration leads to inflation, Pφ < −13ρφ, if the potential energy dominates

over the kinetic energy.

Next, we look in more detail at the evolution of the inflaton φ(t) and the FRW scale factor

a(t). Substituting ρφ from (2.3.27) into the Friedmann equation, H2 = ρφ/(3M2pl), we get

H2 =1

3M2pl

[1

2φ2 + V

]. (F)

Taking a time derivative, we find

2HH =1

3M2pl

[φφ+ V ′φ

], (2.3.29)

where V ′ ≡ dV/dφ. Substituting ρφ and Pφ into the second Friedmann equation (2.2.21),

H = −(ρφ + Pφ)/(2M2pl), we get

H = −1

2

φ2

M2pl

. (2.3.30)

Notice that H is sourced by the kinetic energy density. Combining (2.3.30) with (2.3.29) leads

to the Klein-Gordon equation

φ+ 3Hφ+ V ′ = 0 . (KG)

4You can derive this stress-energy tensor either from Noether’s theorem or from the action of a scalar field.

You will see those derivations in the QFT course: David Tong, Part III Quantum Field Theory.

38 2. Inflation

This is the evolution equation for the scalar field. Notice that the potential acts like a force, V ′,

while the expansion of the universe adds friction, Hφ.

2.3.2 Slow-Roll Inflation

Substituting eq. (2.3.30) into the definition of ε, eq. (2.3.24), we find

ε =12 φ

2

M2plH

2. (2.3.31)

Inflation (ε < 1) therefore occurs if the kinetic energy, 12 φ

2, only makes a small contribution to

the total energy, ρφ = 3M2plH

2. This situation is called slow-roll inflation.

In order for this condition to persist, the acceleration of the scalar field has to be small. To

assess this, it is useful to define the dimensionless acceleration per Hubble time

δ ≡ − φ

Hφ. (2.3.32)

Taking the time-derivative of (2.3.31),

ε =φφ

M2plH

2− φ2H

M2plH

3, (2.3.33)

and comparing to (2.3.25), we find

η =ε

Hε= 2

φ

Hφ− 2

H

H2= 2(ε− δ) . (2.3.34)

Hence, ε, |δ| 1 implies ε, |η| 1.

Slow-roll approximation.—So far, no approximations have been made. We simply noted that in a

regime where ε, |δ| 1, inflation occurs and persists. We now use these conditions to simplify

the equations of motion. This is called the slow-roll approximation. The condition ε 1 implies12 φ

2 V and hence leads to the following simplification of the Friedmann equation (F),

H2 ≈ V

3M2pl

. (FSR)

In the slow-roll approximation, the Hubble expansion is determined completely by the potential

energy. The condition |δ| 1 simplifies the Klein-Gordon equation (KG) to

3Hφ ≈ −V ′ . (KGSR)

This provides a simple relationship between the gradient of the potential and the speed of the

inflaton. Substituting (FSR) and (KGSR) into (2.3.31) gives

ε =12 φ

2

M2plH

2≈M2

pl

2

(V ′

V

)2

. (2.3.35)

Furthermore, taking the time-derivative of (KGSR),

3Hφ+ 3Hφ = −V ′′φ , (2.3.36)

39 2. Inflation

leads to

δ + ε = − φ

Hφ− H

H2≈M2

pl

V ′′

V. (2.3.37)

Hence, a convenient way to assess whether a given potential V (φ) can lead to slow-roll inflation

is to compute the potential slow-roll parameters 5

εv ≡M2

pl

2

(V ′

V

)2

, |ηv| ≡M2pl

|V ′′|V

. (2.3.38)

Successful slow-roll inflation occurs when these parameters are small, εv, |ηv| 1.

Amount of inflation.—The total number of ‘e-folds’ of accelerated expansion are

Ntot ≡∫ aE

aI

d ln a =

∫ tE

tI

H(t) dt , (2.3.39)

where tI and tE are defined as the times when ε(tI) = ε(tE) ≡ 1. In the slow-roll regime, we

can use

Hdt =H

φdφ =

1√2ε

|dφ|Mpl

≈ 1√2εv

|dφ|Mpl

(2.3.40)

to write (2.3.39) as an integral in the field space of the inflaton6

Ntot =

∫ φE

φI

1√2εv

|dφ|Mpl

, (2.3.41)

where φI and φE are defined as the boundaries of the interval where εv < 1. The largest scales

observed in the CMB are produced about 60 e-folds before the end of inflation

Ncmb =

∫ φE

φcmb

1√2εv

|dφ|Mpl

≈ 60 . (2.3.42)

A successful solution to the horizon problem requires Ntot > Ncmb.

Case study: m2φ2 inflation.—As an example, let us give the slow-roll analysis of arguably the simplest

model of inflation: single-field inflation driven by a mass term

V (φ) =1

2m2φ2 . (2.3.43)

The slow-roll parameters are

εv(φ) = ηv(φ) = 2

(Mpl

φ

)2

. (2.3.44)

To satisfy the slow-roll conditions εv, |ηv| < 1, we therefore need to consider super-Planckian values

for the inflaton

φ >√

2Mpl ≡ φE . (2.3.45)

5In contrast, the parameters ε and η are often called the Hubble slow-roll parameters. During slow-roll, the

parameters are related as follows: εv ≈ ε and ηv ≈ 2ε− 12η.

6The absolute value around the integration measure indicates that we pick the overall sign of the integral in

such a way as to make Ntot > 0.

40 2. Inflation

The relation between the inflaton field value and the number of e-folds before the end of inflation is

N(φ) =φ2

4M2pl

− 1

2. (2.3.46)

Fluctuations observed in the CMB are created at

φcmb = 2√NcmbMpl ∼ 15Mpl . (2.3.47)

2.3.3 Reheating∗

During inflation most of the energy density in the universe is in the form of the inflaton po-

tential V (φ). Inflation ends when the potential steepens and the inflaton field picks up kinetic

energy. The energy in the inflaton sector then has to be transferred to the particles of the Stan-

dard Model. This process is called reheating and starts the Hot Big Bang. We will only have

time for a very brief and mostly qualitative description of the absolute basics of the reheating

phenomenon.7 This is non-examinable.

Scalar field oscillations.—After inflation, the inflaton field φ begins to oscillate at the bottom

of the potential V (φ), see fig. 2.5. Assume that the potential can be approximated as V (φ) =12m

2φ2 near the minimum of V (φ), where the amplitude of φ is small. The inflaton is still

homogeneous, φ(t), so its equation of motion is

φ+ 3Hφ = −m2φ . (2.3.48)

The expansion time scale soon becomes much longer than the oscillation period, H−1 m−1.

We can then neglect the friction term, and the field undergoes oscillations with frequency m.

We can write the energy continuity equation as

ρφ + 3Hρφ = −3HPφ = −3

2H(m2φ2 − φ2) . (2.3.49)

The r.h.s. averages to zero over one oscillation period. The oscillating field therefore behaves

like pressureless matter, with ρφ ∝ a−3. The fall in the energy density is reflected in a decrease

of the oscillation amplitude.

Inflaton decay.—To avoid that the universe ends up empty, the inflaton has to couple to Standard

Model fields. The energy stored in the inflaton field will then be transferred into ordinary

particles. If the decay is slow (which is the case if the inflaton can only decay into fermions) the

inflaton energy density follows the equation

ρφ + 3Hρφ = −Γφρφ , (2.3.50)

where Γφ parameterizes the inflaton decay rate. If the inflaton can decay into bosons, the

decay may be very rapid, involving a mechanism called parametric resonance (sourced by Bose

condensation effects). This kind of rapid decay is called preheating, since the bosons thus created

are far from thermal equilibrium.

Thermalisation.—The particles produced by the decay of the inflaton will interact, create other

particles through particle reactions, and the resulting particle soup will eventually reach thermal

7For more details see Baumann, The Physics of Inflation, DAMTP Lecture Notes.

41 2. Inflation

equilibrium with some temperature Trh. This reheating temperature is determined by the energy

density ρrh at the end of the reheating epoch. Necessarily, ρrh < ρφ,E (where ρφ,E is the inflaton

energy density at the end of inflation). If reheating takes a long time, we may have ρrh ρφ,E .

The evolution of the gas of particles into a thermal state can be quite involved. Usually it

is just assumed that it happens eventually, since the particles are able to interact. However,

it is possible that some particles (such as gravitinos) never reach thermal equilibrium, since

their interactions are so weak. In any case, as long as the momenta of the particles are much

higher than their masses, the energy density of the universe behaves like radiation regardless of

the momentum space distribution. After thermalisation of at least the baryons, photons and

neutrinos is complete, the standard Hot Big Bang era begins.

3 Thermal History

In this chapter, we will describe the first three minutes1 in the history of the universe, starting

from the hot and dense state following inflation. At early times, the thermodynamical proper-

ties of the universe were determined by local equilibrium. However, it are the departures from

thermal equilibrium that make life interesting. As we will see, non-equilibrium dynamics allows

massive particles to acquire cosmological abundances and therefore explains why there is some-

thing rather than nothing. Deviations from equilibrium are also crucial for understanding the

origin of the cosmic microwave background and the formation of the light chemical elements.

We will start, in §3.1, with a schematic description of the basic principles that shape the

thermal history of the universe. This provides an overview of the story that will be fleshed out

in much more detail in the rest of the chapter: in §3.2, will present equilibrium thermodynamics

in an expanding universe, while in 3.3, we will introduce the Boltzmann equation and apply it to

several examples of non-equilibrium physics. We will use units in which Boltzmann’s constant

is set equal to unity, kB ≡ 1, so that temperature has units of energy.

3.1 The Hot Big Bang

The key to understanding the thermal history of the universe is the comparison between the

rate of interactions Γ and the rate of expansion H. When Γ H, then the time scale of particle

interactions is much smaller than the characteristic expansion time scale:

tc ≡1

Γ tH ≡

1

H. (3.1.1)

Local thermal equilibrium is then reached before the effect of the expansion becomes relevant.

As the universe cools, the rate of interactions may decrease faster than the expansion rate. At

tc ∼ tH , the particles decouple from the thermal bath. Different particle species may have

different interaction rates and so may decouple at different times.

3.1.1 Local Thermal Equilibrium

Let us first show that the condition (3.1.1) is satisfied for Standard Model processes at temper-

atures above a few hundred GeV. We write the rate of particle interactions as2

Γ ≡ nσv , (3.1.2)

where n is the number density of particles, σ is their interaction cross section, and v is the

average velocity of the particles. For T & 100 GeV, all known particles are ultra-relativistic,

1A wonderful popular account of this part of cosmology is Weinberg’s book The First Three Minutes.2For a process of the form 1 + 2↔ 3 + 4, we would write the interaction rate of species 1 as Γ1 = n2σv, where

n2 is the density of the target species 2 and v is the average relative velocity of 1 and 2. The interaction rate of

species 2 would be Γ2 = n1σv. We have used the expectation that at high energies n1 ∼ n2 ≡ n.

42

43 3. Thermal History

and hence v ∼ 1. Since particle masses can be ignored in this limit, the only dimensionful scale

is the temperature T . Dimensional analysis then gives n ∼ T 3. Interactions are mediated by

gauge bosons, which are massless above the scale of electroweak symmetry breaking. The cross

sections for the strong and electroweak interactions then have a similar dependence, which also

can be estimated using dimensional analysis3

σ ∼

∣∣∣∣∣∣∣∣∣∣2

∼ α2

T 2, (3.1.3)

where α ≡ g2A/4π is the generalized structure constant associated with the gauge boson A. We

find that

Γ = nσv ∼ T 3 × α2

T 2= α2T . (3.1.4)

We wish to compare this to the Hubble rate H ∼ √ρ/Mpl. The same dimensional argument as

before gives ρ ∼ T 4 and hence

H ∼ T 2

M2pl

. (3.1.5)

The ratio of (3.1.4) and (3.1.5) is

Γ

H∼α2Mpl

T∼ 1016 GeV

T, (3.1.6)

where we have used α ∼ 0.01 in the numerical estimate. Below T ∼ 1016 GeV, but above 100

GeV, the condition (3.1.1) is therefore satisfied.

When particles exchange energy and momentum efficiently they reach a state of maximum

entropy. It is a standard result of statistical mechanics that the number of particles per unit

volume in phase space—the distribution function—then takes the form4

f(E) =1

eE/T ± 1, (3.1.7)

where the + sign is for fermions and the − sign for bosons. When the temperature drops below

the mass of the particles, T m, they become non-relativistic and their distribution function

receives an exponential suppression, f → e−m/T . This means that relativistic particles (‘radia-

tion’) dominate the density and pressure of the primordial plasma. The total energy density is

therefore well approximated by summing over all relativistic particles, ρr ∝∑

i

∫d3p fi(p)Ei(p).

The result can be written as (see below)

ρr =π2

30g?(T )T 4 , (3.1.8)

where g?(T ) is the number of relativistic degrees of freedom. Fig. 3.1 shows the evolution of

g?(T ) assuming the particle content of the Standard Model. At early times, all particles are

relativistic and g? = 106.75. The value of g? decreases whenever the temperature of the universe

drops below the mass of a particle species and it becomes non-relativistic. Today, only photons

and (maybe) neutrinos are still relativistic and g? = 3.38.

3Shown in eq. (3.1.3) is the Feynman diagram associated with a 2 → 2 scattering process mediated by the

exchange of a gauge boson. Each vertex contributes a factor of the gauge coupling gA ∝√α. The dependence of

the cross section on α follows from squaring the dependence on α derived from the Feynman diagram, i.e. σ ∝(√α×√α)2 = α2. For more details see the Part III Standard Model course.

4The precise formula will include the chemical potential – see below.


Figure 3.1: Evolution of the number of relativistic degrees of freedom assuming the Standard Model.

3.1.2 Decoupling and Freeze-Out

If equilibrium had persisted until today, the universe would be mostly photons. Any massive

particle species would be exponentially suppressed.5 To understand the world around us, it

is therefore crucial to understand the deviations from equilibrium that led to the freeze-out of

massive particles (see fig. 3.2).

1 10 100

equilibrium

relativistic non-relativistic

freeze-out

relic density

Figure 3.2: A schematic illustration of particle freeze-out. At high temperatures, T m, the particle

abundance tracks its equilibrium value. At low temperatures, T m, the particles freeze out and maintain

a density that is much larger than the Boltzmann-suppressed equilibrium abundance.

Below the scale of electroweak symmetry breaking, T . 100 GeV, the gauge bosons of the

weak interactions, W± and Z, receive masses MW ∼ MZ . The cross section associated with

5This isn’t quite correct for baryons. Since baryon number is a symmetry of the Standard Model, the number

density of baryons can remain significant even in equilibrium.


processes mediated by the weak force becomes

σ ∼

∣∣∣∣∣∣∣∣∣∣∣∣2

∼ G2FT

2 , (3.1.9)

where we have introduced Fermi’s constant,6 GF ∼ α/M2W ∼ 1.17 × 10−5 GeV−2. Notice that

the strength of the weak interactions now decreases as the temperature of the universe drops.

We find thatΓ

H∼α2MplT

3

M4W

∼(

T

1 MeV

)3

, (3.1.10)

which drops below unity at Tdec ∼ 1 MeV. Particles that interact with the primordial plasma

only through the weak interaction therefore decouple around 1 MeV. This decoupling of weak

scale interactions has important consequences for the thermal history of the universe.

3.1.3 A Brief History of the Universe

Table 3.1 lists the key events in the thermal history of the universe:

• Baryogenesis.∗ Relativistic quantum field theory requires the existence of anti-particles

(see Part III Quantum Field Theory). This poses a slight puzzle. Particles and anti-

particles annihilate through processes such as e+ + e− → γ + γ. If initially the universe

was filled with equal amounts of matter and anti-matter then we expect these annihilations

to lead to a universe dominated by radiation. However, we do observe an overabundance

of matter (mostly baryons) over anti-matter in the universe today. Models of baryogenesis

try to derive the observed baryon-to-photon ratio

η ≡ nbnγ∼ 10−9 , (3.1.11)

from some dynamical mechanism, i.e. without assuming a primordial matter-antimatter

asymmetry as an initial condition. Although many ideas for baryogenesis exist, none is

singled out by experimental tests. We will not have much to say about baryogenesis in

this course.

• Electroweak phase transition. At 100 GeV particles receive their masses through the

Higgs mechanism. Above we have seen how this leads to a drastic change in the strength

of the weak interaction.

• QCD phase transition. While quarks are asymptotically free (i.e. weakly interacting)

at high energies, below 150 MeV, the strong interactions between the quarks and the

gluons become important. Quarks and gluons then form bound three-quark systems,

called baryons, and quark-antiquark pairs, called mesons. These baryons and mesons are

the relevant degrees of freedom below the scale of the QCD phase transition.

• Dark matter freeze-out. Since dark matter is very weakly interacting with ordinary

matter we expect it to decouple relatively early on. In §3.3.2, we will study the example

of WIMPs—weakly interacting massive particles that freeze out around 1 MeV. We will

6The 1/M2W comes from the low-momentum limit of the propagator of a massive gauge field.


Event time t redshift z temperature T

Inflation 10−34 s (?) – –

Baryogenesis ? ? ?

EW phase transition 20 ps 1015 100 GeV

QCD phase transition 20 µs 1012 150 MeV

Dark matter freeze-out ? ? ?

Neutrino decoupling 1 s 6× 109 1 MeV

Electron-positron annihilation 6 s 2× 109 500 keV

Big Bang nucleosynthesis 3 min 4× 108 100 keV

Matter-radiation equality 60 kyr 3400 0.75 eV

Recombination 260–380 kyr 1100–1400 0.26–0.33 eV

Photon decoupling 380 kyr 1000–1200 0.23–0.28 eV

Reionization 100–400 Myr 11–30 2.6–7.0 meV

Dark energy-matter equality 9 Gyr 0.4 0.33 meV

Present 13.8 Gyr 0 0.24 meV

Table 3.1: Key events in the thermal history of the universe.

show that choosing natural values for the mass of the dark matter particles and their

interaction cross section with ordinary matter reproduces the observed relic dark matter

density surprisingly well.

• Neutrino decoupling. Neutrinos only interact with the rest of the primordial plasma

through the weak interaction. The estimate in (3.1.10) therefore applies and neutrinos

decouple at 0.8 MeV.

• Electron-positron annihilation. Electrons and positrons annihilate shortly after neu-

trino decoupling. The energies of the electrons and positrons gets transferred to the

photons, but not the neutrinos. In §3.2.4, we will explain that this is the reason why the

photon temperature today is greater than the neutrino temperature.

• Big Bang nucleosynthesis. Around 3 minutes after the Big Bang, the light elements

were formed. In §3.3.4, we will study this process of Big Bang nucleosynthesis (BBN).

• Recombination. Neutral hydrogen forms through the reaction e−+p+ → H+γ when the

temperature has become low enough that the reverse reaction is energetically disfavoured.

We will study recombination in §3.3.3.


• Photon decoupling. Before recombination the strongest coupling between the photons

and the rest of the plasma is through Thomson scattering, e−+γ → e−+γ. The sharp drop

in the free electron density after recombination means that this process becomes inefficient

and the photons decouple. They have since streamed freely through the universe and are

today observed as the cosmic microwave background (CMB).

In the rest of this chapter we will explore in detail where this knowledge about the thermal

history of the universe comes from.

3.2 Equilibrium

3.2.1 Equilibrium Thermodynamics

We have good observational evidence (from the perfect blackbody spectrum of the CMB) that the

early universe was in local thermal equilibrium.7 Moreover, we have seen above that the Standard

Model predicts thermal equilibrium above 100 GeV. To describe this state and the subsequent

evolution of the universe, we need to recall some basic facts of equilibrium thermodynamics,

suitably generalized to apply to an expanding universe.

Microscopic to Macroscopic

Statistical mechanics is the art of turning microscopic laws into an understanding of the macro-

scopic world. I will briefly review this approach for a gas of weakly interacting particles. It is

convenient to describe the system in phase space, where the gas is described by the positions

and momenta of all particles. In quantum mechanics, the momentum eigenstates of a particle

in a volume V = L3 have a discrete spectrum:

The density of states in momentum space p then is L3/h3 = V/h3, and the state density in

phase space x,p is1

h3. (3.2.12)

If the particle has g internal degrees of freedom (e.g. spin), then the density of states becomesg

h3=

g

(2π)3, (3.2.13)

7Strictly speaking, the universe can never truly be in equilibrium since the FRW spacetime doesn’t posses

a time-like Killing vector. But this is physics not mathematics: if the expansion is slow enough, particles have

enough time to settle close to local equilibrium. (And since the universe is homogeneous, the local values of

thermodynamics quantities are also global values.)


where in the second equality we have used natural units with ~ = h/(2π) ≡ 1. To obtain the

number density of a gas of particles we need to know how the particles are distributed amongst

the momentum eigenstates. This information is contained in the (phase space) distribution func-

tion f(x,p, t). Because of homogeneity, the distribution function should, in fact, be independent

of the position x. Moreover, isotropy requires that the momentum dependence is only in terms of

the magnitude of the momentum p ≡ |p|. We will typically leave the time dependence implicit—

it will manifest itself in terms of the temperature dependence of the distribution functions. The

particle density in phase space is then the density of states times the distribution function

g

(2π)3× f(p) . (3.2.14)

The number density of particles (in real space) is found by integrating (3.2.14) over momentum,

n =g

(2π)3

∫d3p f(p) . (3.2.15)

To obtain the energy density of the gas of particles, we have to weight each momentum eigen-

state by its energy. To a good approximation, the particles in the early universe were weakly

interacting. This allows us to ignore the interaction energies between the particles and write the

energy of a particle of mass m and momentum p simply as

E(p) =√m2 + p2 . (3.2.16)

Integrating the product of (3.2.16) and (3.2.14) over momentum then gives the energy density

ρ =g

(2π)3

∫d3p f(p)E(p) . (3.2.17)

Similarly, we define the pressure as

P =g

(2π)3

∫d3p f(p)

p2

3E. (3.2.18)

Pressure.∗—Let me remind you where the p2/3E factor in (3.2.18) comes from. Consider a small area

element of size dA, with unit normal vector n (see fig. 3.3). All particles with velocity |v|, striking

this area element in the time interval between t and t+dt, were located at t = 0 in a spherical shell of

radius R = |v|t and width |v|dt. A solid angle dΩ2 of this shell defines the volume dV = R2|v|dtdΩ2

(see the grey shaded region in fig. 3.3). Multiplying the phase space density (3.2.14) by dV gives the

number of particles in the volume (per unit volume in momentum space) with energy E(|v|),

dN =g

(2π)3f(E)×R2|v|dtdΩ . (3.2.19)

Not all particles in dV reach the target, only those with velocities directed to the area element.

Taking into account the isotropy of the velocity distribution, we find that the total number of

particles striking the area element dA n with velocity v = |v| v is

dNA =|v · n|dA

4πR2× dN =

g

(2π)3f(E)× |v · n|

4πdAdtdΩ , (3.2.20)


where v · n < 0. If these particles are reflected elastically, each transfer momentum 2|p · n| to the

target. Therefore, the contribution of particles with velocity |v| to the pressure is

dP (|v|) =

∫2|p · n|dA dt

dNA =g

(2π)3f(E)× p2

2πE

∫cos2 θ sin θ dθ dφ =

g

(2π)3× f(E)

p2

3E, (3.2.21)

where we have used |v| = |p|/E and integrated over the hemisphere defined by v · n ≡ − cos θ < 0

(i.e. integrating only over particles moving towards dA—see fig. 3.3). Integrating over energy E (or

momentum p), we obtain (3.2.18).

Figure 3.3: Pressure in a weakly interacting gas of particles.

Local Thermal Equilibrium

A system of particles is said to be in kinetic equilibrium if the particles exchange energy and

momentum efficiently. This leads to a state of maximum entropy in which the distribution

functions are given by the Fermi-Dirac and Bose-Einstein distributions8

f(p) =1

e(E(p)−µ)/T ± 1, (3.2.22)

where the + sign is for fermions and the − sign for bosons. At low temperatures, T < E − µ,

both distribution functions reduce to the Maxwell-Boltzmann distribution

f(p) ≈ e−(E(p)−µ)/T . (3.2.23)

The equilibrium distribution functions have two parameters: the temperature T and the chemical

potential µ. The chemical potential may be temperature-dependent. As the universe expands,

T and µ(T ) change in such a way that the continuity equations for the energy density ρ and the

particle number density n are satisfied. Each particle species i (with possibly distinct mi, µi,

Ti) has its own distribution function fi and hence its own ni, ρi, and Pi.

Chemical potential.∗—In thermodynamics, the chemical potential characterizes the response of a

system to a change in particle number. Specifically, it is defined as the derivative of the entropy with

respect to the number of particles, at fixed energy and fixed volume,

µ = −T(∂S

∂N

)U,V

. (3.2.24)

8We use units where Boltzmann’s constant is kB ≡ 1.


The change in entropy of a system therefore is

dS =dU + PdV − µdN

T, (3.2.25)

where µdN is sometimes called the chemical work. A knowledge of the chemical potential of reacting

particles can be used to indicate which way a reaction proceeds. The second law of thermodynamics

means that particles flow to the side of the reaction with the lower total chemical potential. Chemical

equilibrium is reached when the sum of the chemical potentials of the reacting particles is equal to

the sum of the chemical potentials of the products. The rates of the forward and reverse reactions

are then equal.

If a species i is in chemical equilibrium, then its chemical potential µi is related to the chemical

potentials µj of the other species it interacts with. For example, if a species 1 interacts with

species 2, 3 and 4 via the reaction 1 + 2↔ 3 + 4, then chemical equilibrium implies

µ1 + µ2 = µ3 + µ4 . (3.2.26)

Since the number of photons is not conserved (e.g. double Compton scattering e−+γ ↔ e−+γ+γ

happens in equilibrium at high temperatures), we know that

µγ = 0 . (3.2.27)

This implies that if the chemical potential of a particle X is µX , then the chemical potential of

the corresponding anti-particle X is

µX = −µX , (3.2.28)

To see this, just consider particle-antiparticle annihilation, X + X ↔ γ + γ.

Thermal equilibrium is achieved for species which are both in kinetic and chemical equilibrium.

These species then share a common temperature Ti = T .9

3.2.2 Densities and Pressure

Let us now use the results from the previous section to relate the densities and pressure of a gas

of weakly interacting particles to the temperature of the universe.

At early times, the chemical potentials of all particles are so small that they can be neglected.10

Setting the chemical potential to zero, we get

n =g

2π2

∫ ∞0

dpp2

exp[√

p2 +m2/T]± 1

, (3.2.29)

ρ =g

2π2

∫ ∞0

dpp2√p2 +m2

exp[√

p2 +m2/T]± 1

. (3.2.30)

9This temperature is often identified with the photon temperature Tγ — the “temperature of the universe”.10For electrons and protons this is a fact (see Problem Set 2), while for neutrinos it is likely true, but not

proven.


Defining x ≡ m/T and ξ ≡ p/T , this can be written as

n =g

2π2T 3 I±(x) , I±(x) ≡

∫ ∞0

dξξ2

exp[√

ξ2 + x2]± 1

, (3.2.31)

ρ =g

2π2T 4 J±(x) , J±(x) ≡

∫ ∞0

dξξ2√ξ2 + x2

exp[√

ξ2 + x2]± 1

. (3.2.32)

In general, the functions I±(x) and J±(x) have to be evaluated numerically. However, in the

(ultra)relativistic and non-relativistic limits, we can get analytical results.

The following standard integrals will be useful∫ ∞0

dξξn

eξ − 1= ζ(n+ 1) Γ(n+ 1) , (3.2.33)∫ ∞

0dξ ξne−ξ

2= 1

2 Γ(

12(n+ 1)

), (3.2.34)

where ζ(z) is the Riemann zeta-function.

Relativistic Limit

In the limit x→ 0 (m T ), the integral in (3.2.31) reduces to

I±(0) =

∫ ∞0

dξξ2

eξ ± 1. (3.2.35)

For bosons, this takes the form of the integral (3.2.33) with n = 2,

I−(0) = 2ζ(3) , (3.2.36)

where ζ(3) ≈ 1.20205 · · · . To find the corresponding result for fermions, we note that

1

eξ + 1=

1

eξ − 1− 2

e2ξ − 1, (3.2.37)

so that

I+(0) = I−(0)− 2×(

1

2

)3

I−(0) =3

4I−(0) . (3.2.38)

Hence, we get

n =ζ(3)

π2gT 3

1 bosons

34 fermions

. (3.2.39)

A similar computation for the energy density gives

ρ =π2

30gT 4

1 bosons

78 fermions

. (3.2.40)

Relic photons.—Using that the temperature of the cosmic microwave background is T0 = 2.73 K,

show that

nγ,0 =2ζ(3)

π2T 3

0 ≈ 410 photons cm−3 , (3.2.41)

ργ,0 =π2

15T 4

0 ≈ 4.6× 10−34g cm−3 ⇒ Ωγh2 ≈ 2.5× 10−5 . (3.2.42)


Finally, from (3.2.18), it is easy to see that we recover the expected pressure-density relation for

a relativistic gas (i.e. ‘radiation’)

P =1

3ρ . (3.2.43)

Exercise.∗—For µ = 0, the numbers of particles and anti-particles are equal. To find the “net particle

number” let us restore finite µ in the relativistic limit. For fermions with µ 6= 0 and T m, show

that

n− n =g

2π2

∫ ∞0

dp p2

(1

e(p−µ)/T + 1− 1

e(p+µ)/T + 1

)=

1

6π2gT 3

[π2(µT

)+(µT

)3]. (3.2.44)

Note that this result is exact and not a truncated series.

Non-Relativistic Limit

In the limit x 1 (m T ), the integral (3.2.31) is the same for bosons and fermions

I±(x) ≈∫ ∞

0dξ

ξ2

e√ξ2+x2

. (3.2.45)

Most of the contribution to the integral comes from ξ x. We can therefore Taylor expand the

square root in the exponential to lowest order in ξ,

I±(x) ≈∫ ∞

0dξ

ξ2

ex+ξ2/(2x)= e−x

∫ ∞0

dξ ξ2e−ξ2/(2x) = (2x)3/2e−x

∫ ∞0

dξ ξ2e−ξ2. (3.2.46)

The last integral is of the form of the integral (3.2.34) with n = 2. Using Γ(32) =

√π/2, we get

I±(x) =

√π

2x3/2e−x , (3.2.47)

which leads to

n = g

(mT

2π

)3/2

e−m/T . (3.2.48)

As expected, massive particles are exponentially rare at low temperatures, T m. At lowest

order in the non-relativistic limit, we have E(p) ≈ m and the energy density is simply equal to

the mass density

ρ ≈ mn . (3.2.49)

Exercise.—Using E(p) =√m2 + p2 ≈ m+ p2/2m, show that

ρ = mn+3

2nT . (3.2.50)

Finally, from (3.2.18), it is easy to show that a non-relativistic gas of particles acts like pres-

sureless dust (i.e. ‘matter’)

P = nT ρ = mn . (3.2.51)


Exercise.—Derive (3.2.51). Notice that this is nothing but the ideal gas law, PV = NkBT .

By comparing the relativistic limit (T m) and the non-relativistic limit (T m), we see

that the number density, energy density, and pressure of a particle species fall exponentially (are

“Boltzmann suppressed”) as the temperature drops below the mass of the particle. We interpret

this as the annihilation of particles and anti-particles. At higher energies these annihilations

also occur, but they are balanced by particle-antiparticle pair production. At low temperatures,

the thermal particle energies aren’t sufficient for pair production.

Exercise.—Restoring finite µ in the non-relativistic limit, show that

n = g

(mT

2π

)3/2

e−(m−µ)/T , (3.2.52)

n− n = 2g

(mT

2π

)3/2

e−m/T sinh(µT

). (3.2.53)

Effective Number of Relativistic Species

Let T be the temperature of the photon gas. The total radiation density is the sum over the

energy densities of all relativistic species

ρr =∑i

ρi =π2

30g?(T )T 4 , (3.2.54)

where g?(T ) is the effective number of relativistic degrees of freedom at the temperature T . The

sum over particle species may receive two types of contributions:

• Relativistic species in thermal equilibrium with the photons, Ti = T mi,

gth? (T ) =∑i=b

gi +7

8

∑i=f

gi . (3.2.55)

When the temperature drops below the mass mi of a particle species, it becomes non-

relativistic and is removed from the sum in (3.2.55). Away from mass thresholds, the

thermal contribution is independent of temperature.

• Relativistic species that are not in thermal equilibrium with the photons, Ti 6= T mi,

gdec? (T ) =∑i=b

gi

(TiT

)4

+7

8

∑i=f

gi

(TiT

)4

. (3.2.56)

We have allowed for the decoupled species to have different temperatures Ti. This will be

relevant for neutrinos after e+e− annihilation (see §3.2.4).

Fig. 3.4 shows the evolution of g?(T ) assuming the Standard Model particle content (see

table 3.2). At T & 100 GeV, all particles of the Standard Model are relativistic. Adding up


Table 3.2: Particle content of the Standard Model.

type mass spin g

quarks t, t 173 GeV 12 2 · 2 · 3 = 12

b, b 4 GeV

c, c 1 GeV

s, s 100 MeV

d, s 5 MeV

u, u 2 MeV

gluons gi 0 1 8 · 2 = 16

leptons τ± 1777 MeV 12 2 · 2 = 4

µ± 106 MeV

e± 511 keV

ντ , ντ < 0.6 eV 12 2 · 1 = 2

νµ, νµ < 0.6 eV

νe, νe < 0.6 eV

gauge bosons W+ 80 GeV 1 3

W− 80 GeV

Z0 91 GeV

γ 0 2

Higgs boson H0 125 GeV 0 1

their internal degrees of freedom we get:11

gb = 28 photons (2), W± and Z0 (3 · 3), gluons (8 · 2), and Higgs (1)

gf = 90 quarks (6 · 12), charged leptons (3 · 4), and neutrinos (3 · 2)

and hence

g? = gb +7

8gf = 106.75 . (3.2.57)

As the temperature drops, various particle species become non-relativistic and annihilate. To

estimate g? at a temperature T we simply add up the contributions from all relativistic degrees

of freedom (with m T ) and discard the rest.

Being the heaviest particles of the Standard Model, the top quarks annihilates first. At

T ∼ 16mt ∼ 30 GeV,12 the effective number of relativistic species is reduced to g? = 106.75 −

11Here, we have used that massless spin-1 particles (photons and gluons) have two polarizations, massive spin-1

particles (W±, Z) have three polarizations and massive spin- 12

particles (e±, µ±, τ± and quarks) have two spin

states. We assumed that the neutrinos are purely left-handed (i.e. we only counted one helicity state). Also,

remember that fermions have anti-particles.12The transition from relativistic to non-relativistic behaviour isn’t instantaneous. About 80% of the particle-

antiparticle annihilations takes place in the interval T = m→ 16m.


78 × 12 = 96.25. The Higgs boson and the gauge bosons W±, Z0 annihilate next. This happens

roughly at the same time. At T ∼ 10 GeV, we have g? = 96.26 − (1 + 3 · 3) = 86.25. Next,

the bottom quarks annihilate (g? = 86.25 − 78 × 12 = 75.75), followed by the charm quarks

and the tau leptons (g? = 75.75 − 78 × (12 + 4) = 61.75). Before the strange quarks had

time to annihilate, something else happens: matter undergoes the QCD phase transition. At

T ∼ 150 MeV, the quarks combine into baryons (protons, neutrons, ...) and mesons (pions, ...).

There are many different species of baryons and mesons, but all except the pions (π±, π0) are

non-relativistic below the temperature of the QCD phase transition. Thus, the only particle

species left in large numbers are the pions, electrons, muons, neutrinos, and the photons. The

three pions (spin-0) correspond to g = 3 · 1 = 3 internal degrees of freedom. We therefore get

g? = 2 + 3 + 78 × (4 + 4 + 6) = 17.25. Next electrons and positrons annihilate. However, to

understand this process we first need to talk about entropy.

Figure 3.4: Evolution of relativistic degrees of freedom g?(T ) assuming the Standard Model particle content.

The dotted line stands for the number of effective degrees of freedom in entropy g?S(T ).

3.2.3 Conservation of Entropy

To describe the evolution of the universe it is useful to track a conserved quantity. As we will

see, in cosmology entropy is more informative than energy. According to the second law of

thermodynamics, the total entropy of the universe only increases or stays constant. It is easy to

show that the entropy is conserved in equilibrium (see below). Since there are far more photons

than baryons in the universe, the entropy of the universe is dominated by the entropy of the

photon bath (at least as long as the universe is sufficiently uniform). Any entropy production

from non-equilibrium processes is therefore total insignificant relative to the total entropy. To

a good approximation we can therefore treat the expansion of the universe as adiabatic, so that

the total entropy stays constant even beyond equilibrium.


Exercise.—Show that the following holds for particles in equilibrium (which therefore have the cor-

responding distribution functions) and µ = 0:

∂P

∂T=ρ+ P

T. (3.2.58)

Consider the second law of thermodynamics: TdS = dU + PdV . Using U = ρV , we get

dS =1

T

(d[(ρ+ P )V

]− V dP

)=

1

Td[(ρ+ P )V

]− V

T 2(ρ+ P ) dT

= d

[ρ+ P

TV

], (3.2.59)

where we have used (3.2.58) in the second line. To show that entropy is conserved in equilibrium,

we consider

dS

dt=

d

dt

[ρ+ P

TV

]=V

T

[dρ

dt+

1

V

dV

dt(ρ+ P )

]+V

T

[dP

dt− ρ+ P

T

dT

dt

]. (3.2.60)

The first term vanishes by the continuity equation, ρ+3H(ρ+P ) = 0. (Recall that V ∝ a3.) The

second term vanishes by (3.2.58). This established the conservation of entropy in equilibrium.

In the following, it will be convenient to work with the entropy density, s ≡ S/V . From

(3.2.59), we learn that

s =ρ+ P

T. (3.2.61)

Using (3.2.40) and (3.2.51), the total entropy density for a collection of different particle species is

s =∑i

ρi + PiTi

≡ 2π2

45g?S(T )T 3 , (3.2.62)

where we have defined the effective number of degrees of freedom in entropy,

g?S(T ) = gth?S(T ) + gdec?S (T ) . (3.2.63)

Note that for species in thermal equilibrium gth?S(T ) = gth? (T ). However, given that si ∝ T 3i , for

decoupled species we get

gdec?S (T ) ≡∑i=b

gi

(TiT

)3

+7

8

∑i=f

gi

(TiT

)3

6= gdec? (T ) . (3.2.64)

Hence, g?S is equal to g? only when all the relativistic species are in equilibrium at the same

temperature. In the real universe, this is the case until t ≈ 1 sec (cf. fig. 3.4).

The conservation of entropy has two important consequences:


• It implies that s ∝ a−3. The number of particles in a comoving volume is therefore

proportional to the number density ni divided by the entropy density

Ni ≡nis. (3.2.65)

If particles are neither produced nor destroyed, then ni ∝ a−3 and Ni is constant. This is

case, for example, for the total baryon number after baryogenesis, nB/s ≡ (nb − nb)/s.

• It implies, via eq. (3.2.62), that

g?S(T )T 3 a3 = const. , or T ∝ g−1/3?S a−1 . (3.2.66)

Away from particle mass thresholds g?S is approximately constant and T ∝ a−1, as ex-

pected. The factor of g−1/3?S accounts for the fact that whenever a particle species becomes

non-relativistic and disappears, its entropy is transferred to the other relativistic species

still present in the thermal plasma, causing T to decrease slightly less slowly than a−1.

We will see an example in the next section (cf. fig. 3.5).

Substituting T ∝ g−1/3?S a−1 into the Friedmann equation

H =1

a

da

dt'( ρr

3M2pl

)1/2' π

3

( g?10

)1/2 T 2

Mpl, (3.2.67)

we reproduce the usual result for a radiation dominated universe, a ∝ t1/2, except that

there is a change in the scaling every time g?S changes. For T ∝ t−1/2, we can integrate

the Friedmann equation and get the temperature as a function of time

T

1 MeV' 1.5g

−1/4?

(1sec

t

)1/2

. (3.2.68)

It is a useful rule of thumb that the temperature of the universe 1 second after the Big

Bang was about 1 MeV, and evolved as t−1/2 before that.

3.2.4 Neutrino Decoupling

Neutrinos are coupled to the thermal bath via weak interaction processes like

νe + νe ↔ e+ + e− ,

e− + νe ↔ e− + νe .(3.2.69)

The cross section for these interactions was estimated in (3.1.9), σ ∼ G2FT

2, and hence it was

found that Γ ∼ G2FT

5. As the temperature decreases, the interaction rate drops much more

rapidly that the Hubble rate H ∼ T 2/Mpl:

Γ

H∼(

T

1 MeV

)3

. (3.2.70)

We conclude that neutrinos decouple around 1 MeV. (A more accurate computation gives

Tdec ∼ 0.8 MeV.) After decoupling, the neutrinos move freely along geodesics and preserve

to an excellent approximate the relativistic Fermi-Dirac distribution (even after they become

non-relativistic at later times). In §1.2.1, we showed the physical momentum of a particle scales


as p ∝ a−1. It is therefore convenient to define the time-independent combination q ≡ ap, so

that the neutrino number density is

nν ∝ a−3

∫d3q

1

exp(q/aTν) + 1. (3.2.71)

After decoupling, particle number conservation requires nν ∝ a−3. This is only consistent with

(3.2.71) if the neutrino temperature evolves as Tν ∝ a−1. As long as the photon temperature13

Tγ scales in the same way, we still have Tν = Tγ . However, particle annihilations will cause a

deviation from Tγ ∝ a−1 in the photon temperature.

3.2.5 Electron-Positron Annihilation

Shortly after the neutrinos decouple, the temperature drops below the electron mass and electron-

positron annihilation occurs

e+ + e− ↔ γ + γ . (3.2.72)

The energy density and entropy of the electrons and positrons are transferred to the photons,

but not to the decoupled neutrinos. The photons are thus “heated” (the photon temperature

does not decrease as much) relative to the neutrinos (see fig. 3.5). To quantify this effect, we

photon heating

neutrino decoupling

electron-positronannihilation

Figure 3.5: Thermal history through electron-positron annihilation. Neutrinos are decoupled and their

temperature redshifts simply as Tν ∝ a−1. The energy density of the electron-positron pairs is transferred

to the photon gas whose temperature therefore redshifts more slowly, Tγ ∝ g−1/3?S a−1.

consider the change in the effective number of degrees of freedom in entropy. If we neglect

neutrinos and other decoupled species,14 we have

gth?S =

2 + 7

8 × 4 = 112 T & me

2 T < me

. (3.2.73)

Since, in equilibrium, gth?S(aTγ)3 remains constant, we find that aTγ increases after electron-

positron annihilation, T < me, by a factor (11/4)1/3, while aTν remains the same. This means

13For the moment we will restore the subscript on the photon temperature to highlight the difference with the

neutrino temperature.14Obviously, entropy is separately conserved for the thermal bath and the decoupling species.


that the temperature of neutrinos is slightly lower than the photon temperature after e+e−

annihilation,

Tν =

(4

11

)1/3

Tγ . (3.2.74)

For T me, the effective number of relativistic species (in energy density and entropy) there-

fore is

g? = 2 +7

8× 2Neff

(4

11

)4/3

= 3.36 , (3.2.75)

g?S = 2 +7

8× 2Neff

(4

11

)= 3.94 , (3.2.76)

where we have introduced the parameter Neff as the effective number of neutrino species in the

universe. If neutrinos decoupling was instantaneous then we have Neff = 3. However, neutrino

decoupling was not quite complete when e+e− annihilation began, so some of the energy and

entropy did leak to the neutrinos. Taking this into account15 raises the effective number of

neutrinos to Neff = 3.046.16 Using this value in (3.2.75) and (3.2.76) explains the final values of

g?(T ) and g?S(T ) in fig. 3.1.

3.2.6 Cosmic Neutrino Background

The relation (3.2.74) holds until the present. The cosmic neutrino background (CνB) there-

fore has a slightly lower temperature, Tν,0 = 1.95 K = 0.17 meV, than the cosmic microwave

background, T0 = 2.73 K = 0.24 meV. The number density of neutrinos is

nν =3

4Neff ×

4

11nγ . (3.2.77)

Using (3.2.41), we see that this corresponds to 112 neutrinos cm−3 per flavour. The present

energy density of neutrinos depends on whether the neutrinos are relativistic or non-relativistic

today. It used to be believe that neutrinos were massless in which case we would have

ρν =7

8Neff

(4

11

)4/3

ργ ⇒ Ωνh2 ≈ 1.7× 10−5 (mν = 0) . (3.2.78)

Neutrino oscillation experiments have since shown that neutrinos do have mass. The minimum

sum of the neutrino masses is∑mν,i > 60 meV. Massive neutrinos behave as radiation-like

particles in the early universe17, and as matter-like particles in the late universe (see fig. 3.6).

On Problem Set 2, you will show that energy density of massive neutrinos, ρν =∑mν,inν,i,

corresponds to

Ωνh2 ≈

∑mν,i

94 eV. (3.2.79)

By demanding that neutrinos don’t over close the universe, i.e. Ων < 1, one sets a cosmological

upper bound on the sum of the neutrino masses,∑mν,i < 15 eV (using h = 0.7). Measurements

15To get the precise value of Neff one also has to consider the fact that the neutrino spectrum after decoupling

deviates slightly from the Fermi-Dirac distribution. This spectral distortion arises because the energy dependence

of the weak interaction causes neutrinos in the high-energy tail to interact more strongly.16The Planck constraint on Neff is 3.36 ± 0.34. This still leaves room for discovering that Neff 6= 3.046, which

is one of the avenues in which cosmology could discover new physics beyond the Standard Model.17For mν < 0.2 eV, neutrinos are relativistic at recombination.


of tritium β-decay, in fact, find that∑mν,i < 6 eV. Moreover, observations of the cosmic

microwave background, galaxy clustering and type Ia supernovae together put an even stronger

bound,∑mν,i < 1 eV. This implies that although neutrinos contribute at least 25 times the

energy density of photons, they are still a subdominant component overall, 0.001 < Ων < 0.02.

Frac

tiona

l Ene

rgy

Dens

ity

Scale Factor

Temperature [K]

photons

neutrinos

CDM

baryons

Figure 3.6: Evolution of the fractional energy densities of photons, three neutrino species (one massless and

two massive – 0.05 and 0.01 eV), cold dark matter (CDM), baryons, and a cosmological constant (Λ). Notice

the change in the behaviour of the two massive neutrinos when they become non-relativistic particles.

3.3 Beyond Equilibrium

The formal tool to describe the evolution beyond equilibrium is the Boltzmann equation. In

this section, we first introduce the Boltzmann equation and then apply it to three important

examples: (i) the production of dark matter; (ii) the formation of the light elements during Big

Bang nucleosynthesis; and (iii) the recombination of electrons and protons into neutral hydrogen.

3.3.1 Boltzmann Equation

In the absence of interactions, the number density of a particle species i evolves as

dnidt

+ 3a

ani = 0 . (3.3.80)

This is simply a reflection of the fact that the number of particles in a fixed physical volume (V ∝a3) is conserved, so that the density dilutes with the expanding volume, ni ∝ a−3, cf. eq. (1.3.89).

To include the effects of interactions we add a collision term to the r.h.s. of (3.3.80),

1

a3

d(nia3)

dt= Ci[nj] . (3.3.81)

This is the Boltzmann equation. The form of the collision term depends on the specific inter-

actions under consideration. Interactions between three or more particles are very unlikely, so


we can limit ourselves to single-particle decays and two-particle scatterings / annihilations. For

concreteness, let us consider the following process

1 + 2 3 + 4 , (3.3.82)

i.e. particle 1 can annihilate with particle 2 to produce particles 3 and 4, or the inverse process

can produce 1 and 2. This reaction will capture all processes studied in this chapter. Suppose

we are interested in tracking the number density n1 of species 1. Obviously, the rate of change

in the abundance of species 1 is given by the difference between the rates for producing and

eliminating the species. The Boltzmann equation simply formalises this statement,

1

a3

d(n1a3)

dt= −αn1n2 + β n3n4 . (3.3.83)

We understand the r.h.s. as follows: The first term, −αn1n2, describes the destruction of particles

1, while that second term, +β n3n4. Notice that the first term is proportional to n1 and n2 and

the second term is proportional to n3 and n4. The parameter α = 〈σv〉 is the thermally averaged

cross section.18 The second parameter β can be related to α by noting that the collision term

has to vanish in (chemical) equilibrium

β =

(n1n2

n3n4

)eq

α , (3.3.84)

where neqi are the equilibrium number densities we calculated above. We therefore find

1

a3

d(n1a3)

dt= −〈σv〉

[n1n2 −

(n1n2

n3n4

)eq

n3n4

]. (3.3.85)

It is instructive to write this in terms of the number of particles in a comoving volume, as defined

in (3.2.65), Ni ≡ ni/s. This gives

d lnN1

d ln a= −Γ1

H

[1−

(N1N2

N3N4

)eq

N3N4

N1N2

], (3.3.86)

where Γ1 ≡ n2〈σv〉. The r.h.s. of (3.3.86) contains a factor describing the interaction efficiency,

Γ1/H, and a factor characterizing the deviation from equilibrium, [1− · · · ].For Γ1 H, the natural state of the system is chemical equilibrium. Imagine that we start

with N1 N eq1 (while Ni ∼ N eq

i , i = 2, 3, 4). The r.h.s. of (3.3.86) then is negative, particles

of type 1 are destroyed and N1 is reduced towards the equilibrium value N eq1 . Similarly, if

N1 N eq1 , the r.h.s. of (3.3.86) is positive and N1 is driven towards N eq

1 . The same conclusion

applies if several species deviate from their equilibrium values. As long as the interaction rates

are large, the system quickly relaxes to a steady state where the r.h.s. of (3.3.86) vanishes and

the particles assume their equilibrium abundances.

When the reaction rate drops below the Hubble scale, Γ1 < H, the r.h.s. of (3.3.86) gets

suppressed and the comoving density of particles approaches a constant relic density, i.e. N1 =

const. This is illustrated in fig. 3.2. We will see similar types of evolution when we study the

freeze-out of dark matter particles in the early universe (fig. 3.7), neutrons in BBN (fig. 3.9) and

electrons in recombination (fig. 3.8).

18You will learn in the QFT and Standard Model courses how to compute cross sections σ for elementary

processes. In this course, we will simply use dimensional analysis to estimate the few cross sections that we will

need. The cross section may depend on the relative velocity v of particles 1 and 2. The angle brackets in α = 〈σv〉denote an average over v.


3.3.2 Dark Matter Relics

We start with the slightly speculative topic of dark matter freeze-out. I call this speculative

because it requires us to make some assumptions about the nature of the unknown dark matter

particles. For concreteness, we will focus on the hypothesis that the dark matter is a weakly

interacting massive particle (WIMP).

Freeze-Out

WIMPs were in close contact with the rest of the cosmic plasma at high temperatures, but

then experienced freeze-out at a critical temperature Tf . The purpose of this section is to solve

the Boltzmann equation for such a particle, determining the epoch of freeze-out and its relic

abundance.

To get started we have to assume something about the WIMP interactions in the early uni-

verse. We will imagine that a heavy dark matter particle X and its antiparticle X can annihilate

to produce two light (essentially massless) particles ` and ¯,

X + X ↔ `+ ¯ . (3.3.87)

Moreover, we assume that the light particles are tightly coupled to the cosmic plasma,19 so that

throughout they maintain their equilibrium densities, n` = neq` . Finally, we assume that there

is no initial asymmetry between X and X, i.e. nX = nX . The Boltzmann equation (3.3.85) for

the evolution of the number of WIMPs in a comoving volume, NX ≡ nX/s, then is

dNX

dt= −s〈σv〉

[N2X − (N eq

X )2], (3.3.88)

where N eqX ≡ n

eqX /s. Since most of the interesting dynamics will take place when the temperature

is of order the particle mass, T ∼MX , it is convenient to define a new measure of time,

x ≡ MX

T. (3.3.89)

To write the Boltzmann equation in terms of x rather than t, we note that

dx

dt=

d

dt

(MX

T

)= − 1

T

dT

dtx ' Hx , (3.3.90)

where we have assumed that T ∝ a−1 (i.e. g?S ≈ const. ≡ g?S(MX)) for the times relevant to

the freeze-out. We assume radiation domination so that H = H(MX)/x2. Eq. (3.3.88) then

becomes the so-called Riccati equation,

dNX

dx= − λ

x2

[N2X − (N eq

X )2], (3.3.91)

where we have defined

λ ≡ 2π2

45g?S

M3X〈σv〉

H(MX). (3.3.92)

We will treat λ as a constant (which in more fundamental theories of WIMPs is usually a good

approximation). Unfortunately, even for constant λ, there are no analytic solutions to (3.3.91).

Fig. 3.7 shows the result of a numerical solution for two different values of λ. As expected,

19This would be case case, for instance, if ` and ¯ were electrically charged.


1 10 100

Figure 3.7: Abundance of dark matter particles as the temperature drops below the mass.

at very high temperatures, x < 1, we have NX ≈ N eqX ' 1. However, at low temperatures,

x 1, the equilibrium abundance becomes exponentially suppressed, N eqX ∼ e−x. Ultimately,

X-particles will become so rare that they will not be able to find each other fast enough to

maintain the equilibrium abundance. Numerically, we find that freeze-out happens at about

xf ∼ 10. This is when the solution of the Boltzmann equation starts to deviate significantly

from the equilibrium abundance.

The final relic abundance, N∞X ≡ NX(x = ∞), determines the freeze-out density of dark

matter. Let us estimate its magnitude as a function of λ. Well after freeze-out, NX will be

much larger than N eqX (see fig. 3.7). Thus at late times, we can drop N eq

X from the Boltzmann

equation,dNX

dx' −

λN2X

x2(x > xf ) . (3.3.93)

Integrating from xf , to x =∞, we find

1

N∞X− 1

NfX

=λ

xf, (3.3.94)

where NfX ≡ NX(xf ). Typically, Nf

X N∞X (see fig. 3.7), so a simple analytic approximation is

N∞X 'xfλ

. (3.3.95)

Of course, this still depends on the unknown freeze-out time (or temperature) xf . As we see

from fig. 3.7, a good order-of-magnitude estimate is xf ∼ 10. The value of xf isn’t terribly

sensitive to the precise value of λ, namely xf (λ) ∝ | lnλ|.

Exercise.—Estimate xf (λ) from Γ(xf ) = H(xf ).

Eq. (3.3.95) predicts that the freeze-out abundance N∞X decreases as the interaction rate λ

increases. This makes sense intuitively: larger interactions maintain equilibrium longer, deeper

into the Boltzmann-suppressed regime. Since the estimate in (3.3.95) works quite well, we will

use it in the following.


WIMP Miracle∗

It just remains to relate the freeze-out abundance of dark matter relics to the dark matter

density today:

ΩX ≡ρX,0ρcrit,0

=MXnX,03M2

plH20

=MXNX,0s0

3M2plH

20

= MXN∞X

s0

3M2plH

20

. (3.3.96)

where we have used that the number of WIMPs is conserved after freeze-out, i.e. NX,0 = N∞X .

Substituting N∞X = xf/λ and s0 ≡ s(T0), we get

ΩX =H(MX)

M2X

xf〈σv〉

g?S(T0)

g?S(MX)

T 30

3M2plH

20

, (3.3.97)

where we have used (3.3.92) and (3.2.62). Using (3.2.67) for H(MX), gives

ΩX =π

9

xf〈σv〉

(g?(MX)

10

)1/2 g?S(T0)

g?S(MX)

T 30

M3plH

20

. (3.3.98)

Finally, we substitute the measured values of T0 and H0 and use g?S(T0) = 3.91 and g?S(MX) =

g?(MX):

ΩXh2 ∼ 0.1

(xf10

)( 10

g?(MX)

)1/2 10−8 GeV−2

〈σv〉. (3.3.99)

This reproduces the observed dark matter density if√〈σv〉 ∼ 10−4 GeV−1 ∼ 0.1

√GF .

The fact that a thermal relic with a cross section characteristic of the weak interaction gives the

right dark matter abundance is called the WIMP miracle.

3.3.3 Recombination

An important event in the history of the early universe is the formation of the first atoms. At

temperatures above about 1 eV, the universe still consisted of a plasma of free electrons and

nuclei. Photons were tightly coupled to the electrons via Compton scattering, which in turn

strongly interacted with protons via Coulomb scattering. There was very little neutral hydrogen.

When the temperature became low enough, the electrons and nuclei combined to form neutral

atoms (recombination20), and the density of free electrons fell sharply. The photon mean free

path grew rapidly and became longer than the horizon distance. The photons decoupled from the

matter and the universe became transparent. Today, these photons are the cosmic microwave

background.

Saha Equilibrium

Let us start at T > 1 eV, when baryons and photons were still in equilibrium through electro-

magnetic reactions such as

e− + p+ ↔ H + γ . (3.3.100)

20Don’t ask me why this is called recombination; this is the first time electrons and nuclei combined.


Since T < mi, i = e, p,H, we have the following equilibrium abundances

neqi = gi

(miT

2π

)3/2

exp

(µi −mi

T

), (3.3.101)

where µp +µe = µH (recall that µγ = 0). To remove the dependence on the chemical potentials,

we consider the following ratio(nH

nenp

)eq

=gH

gegp

(mH

memp

2π

T

)3/2

e(mp+me−mH)/T . (3.3.102)

In the prefactor, we can use mH ≈ mp, but in the exponential the small difference between mH

and mp +me is crucial: it is the binding energy of hydrogen

BH ≡ mp +me −mH = 13.6 eV . (3.3.103)

The number of internal degrees of freedom are gp = ge = 2 and gH = 4.21 Since, as far as we

know, the universe isn’t electrically charged, we have ne = np. Eq. (3.3.102) therefore becomes(nH

n2e

)eq

=

(2π

meT

)3/2

eBH/T . (3.3.104)

We wish to follow the free electron fraction defined as the ratio

Xe ≡nenb

, (3.3.105)

where nb is the baryon density. We may write the baryon density as

nb = η nγ = η × 2ζ(3)

π2T 3 , (3.3.106)

where η = 5.5× 10−10(Ωbh2/0.020) is the baryon-to-photon ratio. To simplify the discussion, let

us ignore all nuclei other than protons (over 90% (by number) of the nuclei are protons). The

total baryon number density can then be approximated as nb ≈ np + nH = ne + nH and hence

1−Xe

X2e

=nH

n2e

nb . (3.3.107)

Substituting (3.3.104) and (3.3.106), we arrive at the so-called Saha equation,

(1−Xe

X2e

)eq

=2ζ(3)

π2η

(2πT

me

)3/2

eBH/T . (3.3.108)

Fig. 3.8 shows the redshift evolution of the free electron fraction as predicted both by the

Saha approximation (3.3.108) and by a more exact numerical treatment (see below). The Saha

approximation correctly identifies the onset of recombination, but it is clearly insufficient if the

aim is to determine the relic density of electrons after freeze-out.

21The spins of the electron and proton in a hydrogen atom can be aligned or anti-aligned, giving one singlet

state and one triplet state, so gH = 1 + 3 = 4.


BoltzmannSaha

recombination

decouplingCMB

plasma neutral hydrogen

Figure 3.8: Free electron fraction as a function of redshift.

Hydrogen Recombination

Let us define the recombination temperature Trec as the temperature where22 Xe = 10−1

in (3.3.108), i.e. when 90% of the electrons have combined with protons to form hydrogen.

We find

Trec ≈ 0.3 eV ' 3600 K . (3.3.109)

The reason that Trec BH = 13.6 eV is that there are very many photons for each hydrogen

atom, η ∼ 10−9 1. Even when T < BH, the high-energy tail of the photon distribution

contains photons with energy E > BH so that they can ionize a hydrogen atom.

Exercise.—Confirm the estimate in (3.3.109).

Using Trec = T0(1 + zrec), with T0 = 2.7 K, gives the redshift of recombination,

zrec ≈ 1320 . (3.3.110)

Since matter-radiation equality is at zeq ' 3500, we conclude that recombination occurred

in the matter-dominated era. Using a(t) = (t/t0)2/3, we obtain an estimate for the time of

recombination

trec =t0

(1 + zrec)3/2∼ 290 000 yrs . (3.3.111)

Photon Decoupling

Photons are most strongly coupled to the primordial plasma through their interactions with

electrons

e− + γ ↔ e− + γ , (3.3.112)

22There is nothing deep about the choice Xe(Trec) = 10−1. It is as arbitrary as it looks.


with an interaction rate given by

Γγ ≈ neσT , (3.3.113)

where σT ≈ 2× 10−3 MeV−2 is the Thomson cross section. Since Γγ ∝ ne, the interaction rate

decreases as the density of free electrons drops. Photons and electrons decouple roughly when

the interaction rate becomes smaller than the expansion rate,

Γγ(Tdec) ∼ H(Tdec) . (3.3.114)

Writing

Γγ(Tdec) = nbXe(Tdec)σT =2ζ(3)

π2η σT Xe(Tdec)T

3dec , (3.3.115)

H(Tdec) = H0

√Ωm

(TdecT0

)3/2

. (3.3.116)

we get

Xe(Tdec)T3/2dec ∼

π2

2ζ(3)

H0

√Ωm

ησT T3/20

. (3.3.117)

Using the Saha equation for Xe(Tdec), we find

Tdec ∼ 0.27 eV . (3.3.118)

Notice that although Tdec isn’t far from Trec, the ionization fraction decreases significantly be-

tween recombination and decoupling, Xe(Trec) ' 0.1→ Xe(Tdec) ' 0.01. This shows that a large

degree of neutrality is necessary for the universe to become transparent to photon propagation.

Exercise.—Using (3.3.108), confirm the estimate in (3.3.118).

The redshift and time of decoupling are

zdec ∼ 1100 , (3.3.119)

tdec ∼ 380 000 yrs . (3.3.120)

After decoupling the photons stream freely. Observations of the cosmic microwave background

today allow us to probe the conditions at last-scattering.

Electron Freeze-Out∗

In fig. 3.8, we see that a residual ionisation fraction of electrons freezes out when the interactions

in (3.3.100) become inefficient. To follow the free electron fraction after freeze-out, we need to

solve the Boltzmann equation, just as we did for the dark matter freeze-out.

We apply our non-equilibrium master equation (3.3.85) to the reaction (3.3.100). To a rea-

sonably good approximation the neutral hydrogen tracks its equilibrium abundance throughout,

nH ≈ neqH . The Boltzmann equation for the electron density can then be written as

1

a3

d(nea3)

dt= −〈σv〉

[n2e − (neq

e )2]. (3.3.121)


Actually computing the thermally averaged recombination cross section 〈σv〉 from first principles

is quite involved, but a reasonable approximation turns out to be

〈σv〉 ' σT

(BH

T

)1/2

. (3.3.122)

Writing ne = nbXe and using that nba3 = const., we find

dXe

dx= − λ

x2

[X2e − (Xeq

e )2], (3.3.123)

where x ≡ BH/T . We have used the fact that the universe is matter-dominated at recombination

and defined

λ ≡[nb〈σv〉xH

]x=1

= 3.9× 103

(Ωbh

0.03

). (3.3.124)

Exercise.—Derive eq. (3.3.123).

Notice that eq. (3.3.123) is identical to eq. (3.3.91)—the Riccati equation for dark matter freeze-

out. We can therefore immediately write down the electron freeze-out abundance, cf. eq. (3.3.95),

X∞e 'xfλ

= 0.9× 10−3

(xfxrec

)(0.03

Ωbh

). (3.3.125)

Assuming that freeze-out occurs close to the time of recombination, xrec ≈ 45, we capture the

relic electron abundance pretty well (see fig. 3.8).

Exercise.—Using Γe(Tf ) ∼ H(Tf ), show that the freeze-out temperature satisfies

Xe(Tf )Tf =π2

2ζ(3)

H0

√Ωm

ησTT3/20 B

1/2H

. (3.3.126)

Use the Saha equation to show that Tf ∼ 0.25 eV and hence xf ∼ 54.

3.3.4 Big Bang Nucleosynthesis

Let us return to T ∼ 1 MeV. Photons, electron and positrons are in equilibrium. Neutrinos

are about to decouple. Baryons are non-relativistic and therefore much fewer in number than

the relativistic species. Nevertheless, we now want to study what happened to these trace

amounts of baryonic matter. The total number of nucleons stays constant due to baryon number

conservation. This baryon number can be in the form of protons and neutrons or heavier nuclei.

Weak nuclear reactions may convert neutrons and protons into each other and strong nuclear

reactions may build nuclei from them. In this section, I want to show you how the light elements

hydrogen, helium and lithium were synthesised in the Big Bang. I won’t give a complete account

of all of the complicated details of Big Bang Nucleosynthesis (BBN). Instead, the goal of this

section will be more modest: I want to give you a theoretical understanding of a single number:

the ratio of the density of helium to hydrogen,

nHe

nH∼ 1

16. (3.3.127)

Fig. 3.9 summarizes the four steps that will lead us from protons and neutrons to helium.


Step 2: Neutron DecayStep 1:

Neutron Freeze-Out

equilibrium

Frac

tiona

l Abu

ndan

ce

Temperature [MeV]

Step 3: Helium Fusion

Step 0:Equilibrium

Figure 3.9: Numerical results for helium production in the early universe.

Step 0: Equilibrium Abundances

In principle, BBN is a very complicated process involving many coupled Boltzmann equations

to track all the nuclear abundances. In practice, however, two simplifications will make our life

a lot easier:

1. No elements heavier than helium.

Essentially no elements heavier than helium are produced at appreciable levels. So the

only nuclei that we need to track are hydrogen and helium, and their isotopes: deuterium,

tritium, and 3He.

2. Only neutrons and protons above 0.1 MeV.

Above T ≈ 0.1 MeV only free protons and neutrons exist, while other light nuclei haven’t

been formed yet. Therefore, we can first solve for the neutron/proton ratio and then use

this abundance as input for the synthesis of deuterium, helium, etc.

Let us demonstrate that we can indeed restrict our attention to neutrons and protons above

0.1 MeV. In order to do this, we compare the equilibrium abundances of the different nuclei:

• First, we determine the relative abundances of neutrons and protons. In the early universe,

neutrons and protons are coupled by weak interactions, e.g. β-decay and inverse β-decay

n+ νe ↔ p+ + e− ,

n+ e+ ↔ p+ + νe .(3.3.128)

Let us assume that the chemical potentials of electrons and neutrinos are negligibly small,


so that µn = µp. Using (3.3.101) for neqi , we then have(

nnnp

)eq

=

(mn

mp

)3/2

e−(mn−mp)/T . (3.3.129)

The small difference between the proton and neutron mass can be ignored in the first

factor, but crucially has to be kept in the exponential. Hence, we find(nnnp

)eq

= e−Q/T , (3.3.130)

where Q ≡ mn −mp = 1.30 MeV. For T 1 MeV, there are therefore as many neutrons

as protons. However, for T < 1 MeV, the neutron fraction gets smaller. If the weak

interactions would operate efficiently enough to maintain equilibrium indefinitely, then the

neutron abundance would drop to zero. Luckily, in the real world the weak interactions

are not so efficient.

• Next, we consider deuterium (an isotope of hydrogen with one proton and one neutron).

This is produced in the following reaction

n+ p+ ↔ D + γ . (3.3.131)

Since µγ = 0, we have µn+µp = µD. To remove the dependence on the chemical potentials

we consider (nD

nnnp

)eq

=3

4

(mD

mnmp

2π

T

)3/2

e−(mD−mn−mp)/T , (3.3.132)

where, as before, we have used (3.3.101) for neqi (with gD = 3 and gp = gn = 2). In the

prefactor, mD can be set equal to 2mn ≈ 2mp ≈ 1.9 GeV, but in the exponential the small

difference between mn +mp and mD is crucial: it is the binding energy of deuterium

BD ≡ mn +mp −mD = 2.22 MeV . (3.3.133)

Therefore, as long as chemical equilibrium holds the deuterium-to-proton ratio is(nD

np

)eq

=3

4neqn

(4π

mpT

)3/2

eBD/T . (3.3.134)

To get an order of magnitude estimate, we approximate the neutron density by the baryon

density and write this in terms of the photon temperature and the baryon-to-photon ratio,

nn ∼ nb = η nγ = η × 2ζ(3)

π2T 3 . (3.3.135)

Eq. (3.3.134) then becomes (nD

np

)eq

≈ η(T

mp

)3/2

eBD/T . (3.3.136)

The smallness of the baryon-to-photon ratio η inhibits the production of deuterium until

the temperature drops well beneath the binding energy BD. The temperature has to drop

enough so that eBD/T can compete with η ∼ 10−9. The same applies to all other nuclei. At

temperatures above 0.1 MeV, then, virtually all baryons are in the form of neutrons and

protons. Around this time, deuterium and helium are produced, but the reaction rates are

by now too low to produce any heavier elements.


Step 1: Neutron Freeze-Out

The primordial ratio of neutrons to protons is of particular importance to the outcome of BBN,

since essentially all the neutrons become incorporated into 4He. As we have seen, weak inter-

actions keep neutrons and protons in equilibrium until T ∼ MeV. After that, we must solve

the Boltzmann equation (3.3.85) to track the neutron abundance. Since this is a bit involved, I

won’t describe it in detail (but see the box below). Instead, we will estimate the answer a bit

less rigorously.

It is convenient to define the neutron fraction as

Xn ≡nn

nn + np. (3.3.137)

From the equilibrium ratio of neutrons to protons (3.3.130), we then get

Xeqn (T ) =

e−Q/T

1 + e−Q/T. (3.3.138)

Neutrons follows this equilibrium abundance until neutrinos decouple at23 Tf ∼ Tdec ∼ 0.8 MeV

(see §3.2.4). At this moment, weak interaction processes such as (3.3.128) effectively shut off.

The equilibrium abundance at that time is

Xeqn (0.8 MeV) = 0.17 . (3.3.139)

We will take this as a rough estimate for the final freeze-out abundance,

X∞n ∼ Xeqn (0.8 MeV) ∼ 1

6. (3.3.140)

We have converted the result to a fraction to indicate that this is only an order of magnitude

estimate.

Exact treatment∗.—OK, since you asked, I will show you some details of the more exact treatment.

To be clear, this box is definitely not examinable!

Using the Boltzmann equation (3.3.85), with 1 = neutron, 3 = proton, and 2, 4 = leptons (with

n` = neq` ), we find

1

a3

d(nna3)

dt= −Γn

[nn −

(nnnp

)eq

np

], (3.3.141)

where we have defined the rate for neutron/proton conversion as Γn ≡ n`〈σv〉. Substituting (3.3.137)

and (3.3.138), we finddXn

dt= −Γn

[Xn − (1−Xn)e−Q/T

]. (3.3.142)

Instead of trying to solve this for Xn as a function of time, we introduce a new evolution variable

x ≡ QT. (3.3.143)

We write the l.h.s. of (3.3.142) as

dXn

dt=dx

dt

dXn

dx= − x

T

dT

dt

dXn

dx= xH

dXn

dx, (3.3.144)

23If is fortunate that Tf ∼ Q. This seems to be a coincidence: Q is determined by the strong and electromagnetic

interactions, while the value of Tf is fixed by the weak interaction. Imagine a world in which Tf Q!


where in the last equality we used that T ∝ a−1. During BBN, we have

H =

√ρ

3M2pl

=π

3

√g?10

Q2

Mpl︸︷︷︸≡H1≈ 1.13s−1

1

x2, with g? = 10.75 . (3.3.145)

Eq. (3.3.142) then becomesdXn

dx=

ΓnH1

x[e−x −Xn(1 + e−x)

]. (3.3.146)

Finally, we need an expression for the neutron-proton conversion rate, Γn. You can find a sketch of

the required QFT calculation in Dodelson’s book. Here, I just cite the answer

Γn(x) =255

τn· 12 + 6x+ x2

x5, (3.3.147)

where τn = 886.7 ± 0.8 sec is the neutron lifetime. One can see that the conversion time Γ−1n is

comparable to the age of the universe at a temperature of ∼ 1 MeV. At later times, T ∝ t−1/2 and

Γn ∝ T 3 ∝ t−3/2, so the neutron-proton conversion time Γ−1n ∝ t3/2 becomes longer than the age of

the universe. Therefore we get freeze-out, i.e. the reaction rates become slow and the neutron/proton

ratio approaches a constant. Indeed, solving eq. (3.3.146) numerically, we find (see fig. 3.9)

X∞n ≡ Xn(x =∞) = 0.15 . (3.3.148)

Step 2: Neutron Decay

At temperatures below 0.2 MeV (or t & 100 sec) the finite lifetime of the neutron becomes

important. To include neutron decay in our computation we simply multiply the freeze-out

abundance (3.3.148) by an exponential decay factor

Xn(t) = X∞n e−t/τn =

1

6e−t/τn , (3.3.149)

where τn = 886.7± 0.8 sec.

Step 3: Helium Fusion

At this point, the universe is mostly protons and neutron. Helium cannot form directly because

the density is too low and the time available is too short for reactions involving three or more

incoming nuclei to occur at any appreciable rate. The heavier nuclei therefore have to be built

sequentially from lighter nuclei in two-particle reactions. The first nucleus to form is therefore

deuterium,

n+ p+ ↔ D + γ . (3.3.150)

Only when deuterium is available can helium be formed,

D + p+ ↔ 3He + γ , (3.3.151)

D + 3He ↔ 4He + p+ . (3.3.152)

Since deuterium is formed directly from neutrons and protons it can follow its equilibrium

abundance as long as enough free neutrons are available. However, since the deuterium binding


energy is rather small, the deuterium abundance becomes large rather late (at T < 100 keV).

So although heavier nuclei have larger binding energies and hence would have larger equilibrium

abundances, they cannot be formed until sufficient deuterium has become available. This is the

deuterium bottleneck. Only when there is enough deuterium, can helium be produced. To get

a rough estimate for the time of nucleosynthesis, we determine the temperature Tnuc when the

deuterium fraction in equilibrium would be of order one, i.e. (nD/np)eq ∼ 1. Using (3.3.136), I

find

Tnuc ∼ 0.06 MeV , (3.3.153)

which via (3.2.68) with g? = 3.38 translates into

tnuc = 120 sec

(0.1MeV

Tnuc

)2

∼ 330 sec. (3.3.154)

Comment.—From fig. 3.9, we see that a better estimate would be neqD (Tnuc) ' 10−3neq

p (Tnuc). This

gives Tnuc ' 0.07 MeV and tnuc ' 250 sec. Notice that tnuc τn, so eq. (3.3.149) won’t be very

sensitive to the estimate for tnuc.

Substituting tnuc ∼ 330 sec into (3.3.149), we find

Xn(tnuc) ∼1

8. (3.3.155)

Since the binding energy of helium is larger than that of deuterium, the Boltzmann factor

eB/T favours helium over deuterium. Indeed, in fig. 3.9 we see that helium is produced almost

immediately after deuterium. Virtually all remaining neutrons at t ∼ tnuc then are processed

into 4He. Since two neutrons go into one nucleus of 4He, the final 4He abundance is equal to

half of the neutron abundance at tnuc, i.e. nHe = 12nn(tnuc), or

nHe

nH=nHe

np'

12Xn(tnuc)

1−Xn(tnuc)∼ 1

2Xn(tnuc) ∼

1

16, (3.3.156)

as we wished to show. Sometimes, the result is expressed as the mass fraction of helium,

4nHe

nH∼ 1

4. (3.3.157)

This prediction is consistent with the observed helium in the universe (see fig. 3.10).

BBN as a Probe of BSM Physics

We have arrived at a number for the final helium mass fraction, but we should remember that

this number depends on several input parameters:

• g?: the number of relativistic degrees of freedom determines the Hubble parameter during

the radiation era, H ∝ g1/2? , and hence affects the freeze-out temperature

G2FT

5f ∼

√GNg? T

2f → Tf ∝ g

1/6? . (3.3.158)

Increasing g? increases Tf , which increases the n/p ratio at freeze-out and hence increases

the final helium abundance.


WMAP

4He

3He

D

Mas

s Fra

ctio

n

7Li

WMAP

Num

ber r

elativ

e to

H

Figure 3.10: Theoretical predictions (colored bands) and observational constraints (grey bands).

• τn: a large neutron lifetime would reduce the amount of neutron decay after freeze-out

and therefore would increase the final helium abundance.

• Q: a larger mass difference between neutrons and protons would decrease the n/p ratio

at freeze-out and therefore would decrease the final helium abundance.

• η: the amount of helium increases with increasing η as nucleosythesis starts earlier for

larger baryon density.

• GN : increasing the strength of gravity would increase the freeze-out temperature, Tf ∝G

1/6N , and hence would increase the final helium abundance.

• GF : increasing the weak force would decrease the freeze-out temperature, Tf ∝ G−2/3F ,

and hence would decrease the final helium abundance.

Changing the input, e.g. by new physics beyond the Standard Model (BSM) in the early universe,

would change the predictions of BBN. In this way BBN is a probe of fundamental physics.

Light Element Synthesis∗

To determine the abundances of other light elements, the coupled Boltzmann equations have to

be solved numerically (see fig. 3.11 for the result of such a computation). Fig. 3.10 shows that

theoretical predictions for the light element abundances as a function of η (or Ωb). The fact that

we find reasonably good quantitative agreement with observations is one of the great triumphs

of the Big Bang model.


4HeD

np

7Li

7Be

3He

3HM

ass F

ract

ion

Time [min]

Figure 3.11: Numerical results for the evolution of light element abundances.

The shape of the curves in fig. 3.11 can easily be understood: The abundance of 4He increases

with increasing η as nucleosythesis starts earlier for larger baryon density. D and 3He are burnt

by fusion, thus their abundances decrease as η increases. Finally, 7Li is destroyed by protons at

low η with an efficiency that increases with η. On the other hand, its precursor 7Be is produced

more efficiently as η increases. This explains the valley in the curve for 7Li.

Part II

The Inhomogeneous Universe

76

4 Cosmological Perturbation Theory

So far, we have treated the universe as perfectly homogeneous. To understand the formation

and evolution of large-scale structures, we have to introduce inhomogeneities. As long as these

perturbations remain relatively small, we can treat them in perturbation theory. In particular,

we can expand the Einstein equations order-by-order in perturbations to the metric and the

stress tensor. This makes the complicated system of coupled PDEs manageable.

4.1 Newtonian Perturbation Theory

Newtonian gravity is an adequate description of general relativity on scales well inside the Hubble

radius and for non-relativistic matter (e.g. cold dark matter and baryons after decoupling). We

will start with Newtonian perturbation theory because it is more intuitive than the full treatment

in GR.

4.1.1 Perturbed Fluid Equations

Consider a non-relativistic fluid with mass density ρ, pressure P ρ and velocity u. Denote

the position vector of a fluid element by r and time by t. The equations of motion are given by

basic fluid dynamics.1 Mass conservation implies the continuity equation

∂tρ = −∇r ·(ρu) , (4.1.1)

while momentum conservation leads to the Euler equation

(∂t + u ·∇r)u = −∇rP

ρ−∇rΦ . (4.1.2)

The last equation is simply “F = ma” for a fluid element. The gravitational potential Φ is

determined by the Poisson equation

∇2rΦ = 4πGρ . (4.1.3)

Convective derivative.∗—Notice that the acceleration in (4.1.2) is not given by ∂tu (which mea-

sures how the velocity changes at a given position), but by the “convective time derivative”

Dtu ≡ (∂t + u ·∇)u which follows the fluid element as it moves. Let me remind you how this

comes about.

Consider a fixed volume in space. The total mass in the volume can only change if there is a

flux of momentum through the surface. Locally, this is what the continuity equation describes:

∂tρ + ∇j(ρuj) = 0. Similarly, in the absence of any forces, the total momentum in the volume

1See Landau and Lifshitz, Fluid Mechanics.

77

78 4. Cosmological Perturbation Theory

can only change if there is a flux through the surface: ∂t(ρui) + ∇j(ρuiuj) = 0. Expanding the

derivatives, we get

∂t(ρui) +∇j(ρuiuj) = ρ [∂t + uj∇j ]ui + ui [∂tρ+∇j(ρuj)]︸︷︷︸=0

= ρ [∂t + uj∇j ]ui .

In the absence of forces it is therefore the convective derivative of the velocity, Dtu, that vanishes,

not ∂tu. Adding forces gives the Euler equation.

We wish to see what these equation imply for the evolution of small perturbations around

a homogeneous background. We therefore decompose all quantities into background values

(denoted by an overbar) and perturbations—e.g. ρ(t, r) = ρ(t) + δρ(t, r), and similarly for the

pressure, the velocity and the gravitational potential. Assuming that the fluctuations are small,

we can linearise eqs. (4.1.1) and (4.1.2), i.e. we can drop products of fluctuations.

Static space without gravity

Let us first consider static space and ignore gravity (Φ ≡ 0). It is easy to see that a solution for

the background is ρ = const., P = const. and u = 0. The linearised evolution equations for the

fluctuations are

∂tδρ = −∇r ·(ρu) , (4.1.4)

ρ ∂tu = −∇rδP . (4.1.5)

Combining ∂t (4.1.4) and ∇r·(4.1.5), one finds

∂2t δρ−∇2

r δP = 0 . (4.1.6)

For adiabatic fluctuations (see below), the pressure fluctuations are proportional to the density

fluctuations, δP = c2s δρ, where cs is called the speed of sound. Eq. (4.1.6) then takes the form

of a wave equation (∂2t − c2

s∇2)δρ = 0 . (4.1.7)

This is solved by a plane wave, δρ = A exp[i(ωt − k · r)], where ω = csk, with k ≡ |k|. We see

that in a static spacetime fluctuations oscillate with constant amplitude if we ignore gravity.

Fourier space.—The more formal way to solve PDEs like (4.1.7) is to expand δρ in terms of its Fourier

components

δρ(t, r) =

∫d3k

(2π)3e−ik·rδρk(t) . (4.1.8)

The PDE (4.1.7) turns into an ODE for each Fourier mode(∂2t + c2sk

2)δρk = 0 , (4.1.9)

which has the solution

δρk = Akeiωkt +Bke

−iωkt , ωk ≡ csk . (4.1.10)


Static space with gravity

Now we turn on gravity. Eq. (4.1.7) then gets a source term(∂2t − c2

s∇2r

)δρ = 4πGρδρ , (4.1.11)

where we have used the perturbed Poisson equation, ∇2δΦ = 4πGδρ. This is still solved by

δρ = A exp[i(ωt− k · r)], but now with

ω2 = c2sk

2 − 4πGρ . (4.1.12)

We see that there is a critical wavenumber for which the frequency of oscillations is zero:

kJ ≡√

4πGρ

cs. (4.1.13)

For small scales (i.e. large wavenumber), k > kJ, the pressure dominates and we find the same

oscillations as before. However, on large scales, k < kJ, gravity dominates, the frequency ω

becomes imaginary and the fluctuations grow exponentially. The crossover happens at the Jeans’

length

λJ =2π

kJ= cs

√π

Gρ. (4.1.14)

Expanding space

In an expanding space, we have the usual relationship between physical coordinates r and

comoving coordinates x,

r(t) = a(t)x . (4.1.15)

The velocity field is then given by

u(t) = r = Hr + v , (4.1.16)

where Hr is the Hubble flow and v = ax is the proper velocity. In a static spacetime, the time

and space derivates defined from t and r were independent. In an expanding spacetime this is

not the case anymore. It is then convenient to use space derivatives defined with respect to the

comoving coordinates x, which we denote by ∇x. Using (4.1.15), we have

∇r = a−1∇x . (4.1.17)

The relationship between time derivatives at fixed r and at fixed x is(∂

∂t

)r

=

(∂

∂t

)x

+

(∂x

∂t

)r

·∇x =

(∂

∂t

)x

+

(∂a−1(t)r

∂t

)r

·∇x

=

(∂

∂t

)x

−Hx ·∇x . (4.1.18)

From now on, we will drop the subscripts x.


With this in mind, let us look at the fluid equations in an expanding universe:

• Continuity equation

Substituting (4.1.17) and (4.1.18) for ∇r and ∂t in the continuity equation (4.1.1), we get[∂

∂t−Hx ·∇

] [ρ(1 + δ)

]+

1

a∇ ·

[ρ(1 + δ)(Hax + v)

]= 0 , (4.1.19)

Here, I have introduced the fractional density perturbation

δ ≡ δρ

ρ. (4.1.20)

Sometimes δ is called the density contrast.

Let us analyse this order-by-order in perturbation theory:

– At zeroth order in fluctuations (i.e. dropping the perturbations δ and v), we have

∂ρ

∂t+ 3Hρ = 0 , (4.1.21)

where I have used ∇x ·x = 3. We recognise this as the continuity equation for the

homogeneous mass density, ρ ∝ a−3.

– At first order in fluctuations (i.e. dropping products of δ and v), we get[∂

∂t−Hx ·∇

] [ρδ]

+1

a∇ ·

[ρHaxδ + ρv

]= 0 , (4.1.22)

which we can write as [∂ρ

∂t+ 3Hρ

]δ + ρ

∂δ

∂t+ρ

a∇ · v = 0 . (4.1.23)

The first term vanishes by (4.1.21), so we find

δ = −1

a∇ · v , (4.1.24)

where we have used an overdot to denote the derivative with respect to time.

• Euler equation

Similar manipulations of the Euler equation (4.1.2) lead to

v +Hv = − 1

aρ∇δP − 1

a∇δΦ . (4.1.25)

In the absence of pressure and gravitational perturbations, this equation simply says that

v ∝ a−1, which is something we already discovered in Chapter 1.

• Poisson equation

It takes hardly any work to show that the Poisson equation (4.1.3) becomes

∇2δΦ = 4πGa2ρδ . (4.1.26)


Exercise.—Derive eq. (4.1.25).

4.1.2 Jeans’ Instability

Combining ∂t (4.1.24) with ∇·(4.1.25) and (4.1.26), we find

δ + 2Hδ − c2s

a2∇2δ = 4πGρδ . (4.1.27)

This implies the same Jeans’ length as in (4.1.14), but unlike the case of a static spacetime, it

now depends on time via ρ(t) and cs(t). Compared to (4.1.11), the equation of motion in the

expanding spacetime includes a friction term, 2Hδ. This has two effects: Below the Jeans’ length,

the fluctuations oscillate with decreasing amplitude. Above the Jeans’ length, the fluctuations

experience power-law growth, rather than the exponential growth we found for static space.

4.1.3 Dark Matter inside Hubble

The Newtonian framework describes the evolution of matter fluctuations. We can apply it to

the evolution dark matter on sub-Hubble scales. (We will ignore small effects due to baryons.)

• During the matter-dominated era, eq. (4.1.27) reads

δm + 2Hδm − 4πGρmδm = 0 , (4.1.28)

where we have dropped the pressure term, since cs = 0 for linearised CDM fluctuations.

(Non-linear effect produce a finite, but small, sound speed.) Since a ∝ t2/3, we have

H = 2/3t and hence

δm +4

3tδm −

2

3t2δm = 0 , (4.1.29)

where we have used 4πGρm = 32H

2. Trying δm ∝ tp gives the following two solutions:

δm ∝

t−1 ∝ a−3/2

t2/3 ∝ a. (4.1.30)

Hence, the growing mode of dark matter fluctuations grows like the scale factor during the

MD era. This is a famous result that is worth remembering.

• During the radiation-dominated era, eq. (4.1.27) gets modified to

δm + 2Hδm − 4πG∑I

ρI δI = 0 , (4.1.31)

where the sum is over matter and radiation. (It is the total density fluctuation δρ = δρm+

δρr which sources δΦ!) Radiation fluctuations on scales smaller than the Hubble radius

oscillate as sound waves (supported by large radiation pressure) and their time-averaged

density contrast vanishes. To prove this rigorously requires relativistic perturbation theory

(see below). It follows that the CDM is essentially the only clustered component during

the acoustic oscillations of the radiation, and so

δm +1

tδm − 4πGρmδm ≈ 0 . (4.1.32)


Since δm evolves only on cosmological timescales (it has no pressure support for it to do

otherwise), we have

δm ∼ H2δm ∼8πG

3ρrδm 4πGρmδm , (4.1.33)

where we have used that ρr ρm. We can therefore ignore the last term in (4.1.32)

compared to the others. We then find

δm ∝

const.

ln t ∝ ln a. (4.1.34)

We see that the rapid expansion due to the effectively unclustered radiation reduces the

growth of δm to only logarithmic. This is another fact worth remembering: we need to

wait until the universe becomes matter dominated in order for the dark matter density

fluctuations to grow significantly.

• During the Λ-dominated era, eq. (4.1.27) reads

δm + 2Hδm − 4πG∑i

ρI δI = 0 , (4.1.35)

where I = m,Λ. As far as we can tell, dark energy doesn’t cluster (almost by definition),

so we can write

δm + 2Hδm − 4πGρm δm = 0 , (4.1.36)

Notice that this is not the same as (4.1.28), because H is different. Indeed, in the Λ-

dominated regime H2 ≈ const. 4πGρm. Dropping the last term in (4.1.36), we get

δm + 2Hδm ≈ 0 , (4.1.37)

which has the following solutions

δm ∝

const.

e−2Ht ∝ a−2. (4.1.38)

We see that the matter fluctuations stop growing once dark energy comes to dominate.

4.2 Relativistic Perturbation Theory

The Newtonian treatment of cosmological perturbations is inadequate on scales larger than the

Hubble radius, and for relativistic fluids (like photons and neutrinos). The correct description

requires a full general-relativistic treatment which we will now develop.

4.2.1 Perturbed Spacetime

The basic idea is to consider small perturbations δgµν around the FRW metric gµν ,

gµν = gµν + δgµν . (4.2.39)

Through the Einstein equations, the metric perturbations will be coupled to perturbations in

the matter distribution.


Perturbations of the Metric

To avoid unnecessary technical distractions, we will only present the case of a flat FRW back-

ground spacetime

ds2 = a2(τ)[dτ2 − δijdxidxj

]. (4.2.40)

The perturbed metric can then be written as

ds2 = a2(τ)[(1 + 2A)dτ2 − 2Bidx

idτ − (δij + hij)dxidxj

], (4.2.41)

where A, Bi and hij are functions of space and time. We shall adopt the useful convention that

Latin indices on spatial vectors and tensors are raised and lowered with δij , e.g. hii = δijhij .

Scalar, Vectors and Tensors

It will be extremely useful to perform a scalar-vector-tensor (SVT) decomposition of the pertur-

bations. For 3-vectors, this should be familiar. It simply means that we can split any 3-vector

into the gradient of a scalar and a divergenceless vector

Bi = ∂iB︸︷︷︸scalar

+ Bi︸︷︷︸vector

, (4.2.42)

with ∂iBi = 0. Similarly, any rank-2 symmetric tensor can be written

hij = 2Cδij + 2∂〈i∂j〉E︸︷︷︸scalar

+ 2∂(iEj)︸︷︷︸vector

+ 2Eij︸︷︷︸tensor

, (4.2.43)

where

∂〈i∂j〉E ≡(∂i∂j −

1

3δij∇2

)E , (4.2.44)

∂(iEj) ≡1

2

(∂iEj + ∂jEi

). (4.2.45)

As before, the hatted quantities are divergenceless, i.e. ∂iEi = 0 and ∂iEij = 0. The tensor

perturbation is traceless, Eii = 0. The 10 degrees of freedom of the metric have thus been

decomposed into 4 + 4 + 2 SVT degrees of freedom:

• scalars: A, B, C, E

• vectors: Bi, Ei

• tensors: Eij

What makes the SVT-decomposition so powerful is the fact that the Einstein equations for

scalars, vectors and tensors don’t mix at linear order and can therefore be treated separately.

In these lectures, we will mostly be interested in scalar fluctuations and the associated density

perturbations. Vector perturbations aren’t produced by inflation and even if they were, they

would decay quickly with the expansion of the universe. Tensor perturbations are an important

prediction of inflation and we will discuss them briefly in Chapter 6.


The Gauge Problem

Before we continue, we have to address an important subtlety. The metric perturbations

in (4.2.41) aren’t uniquely defined, but depend on our choice of coordinates or the gauge choice.

In particular, when we wrote down the perturbed metric, we implicitly chose a specific time

slicing of the spacetime and defined specfic spatial coordinates on these time slices. Making a

different choice of coordinates, can change the values of the perturbation variables. It may even

introduce fictitious perturbations. These are fake perturbations that can arise by an inconvenient

choice of coordinates even if the background is perfectly homogeneous.

For example, consider the homogeneous FRW spacetime (4.2.40) and make the following

change of the spatial coordinates, xi 7→ xi = xi + ξi(τ,x). We assume that ξi is small, so that it

can also be treated as a perturbation. Using dxi = dxi−∂τξidτ −∂kξidxk, eq. (4.2.40) becomes

ds2 = a2(τ)[dτ2 − 2ξ ′i dx

idτ −(δij + 2∂(iξj)

)dxidxj

], (4.2.46)

where we have dropped terms that are quadratic in ξi and defined ξ ′i ≡ ∂τξi. We apparently

have introduced the metric perturbations Bi = ξ′i and Ei = ξi. But these are just fictitious

gauge modes that can be removed by going back to the old coordinates.

Similar, we can change our time slicing, τ 7→ τ + ξ0(τ,x). The homogeneous density of the

universe then gets perturbed, ρ(τ) 7→ ρ(τ + ξ0(τ,x)) = ρ(τ) + ρ ′ξ0. So even in an unperturbed

universe, a change of the time coordinate can introduce a fictitious density perturbation

δρ = ρ ′ξ0 . (4.2.47)

Similarly, we can remove a real perturbation in the energy density by choosing the hypersurface

of constant time to coincide with the hypersurface of constant energy density. Then δρ = 0

although there are real inhomogeneities.

These examples illustrate that we need a more physical way to identify true perturbations.

One way to do this is to define perturbations in such a way that they don’t change under a

change of coordinates.

Gauge Transformations

Consider the coordinate transformation

Xµ 7→ Xµ ≡ Xµ + ξµ(τ,x) , where ξ0 ≡ T , ξi ≡ Li = ∂iL+ Li . (4.2.48)

We have split the spatial shift Li into a scalar, L, and a divergenceless vector, Li. We wish to

know how the metric transforms under this change of coordinates. The trick is to exploit the

invariance of the spacetime interval,

ds2 = gµν(X)dXµdXν = gαβ(X)dXαdXβ , (4.2.49)

where I have used a different set of dummy indices on both sides to make the next few lines

clearer. Writing dXα = (∂Xα/∂Xµ)dXµ (and similarly for dXβ), we find

gµν(X) =∂Xα

∂Xµ

∂Xβ

∂Xνgαβ(X) . (4.2.50)

This relates the metric in the old coordinates, gµν , to the metric in the new coordinates, gαβ.


Let us see what (4.2.50) implies for the transformation of the metric perturbations in (4.2.41).

I will work out the 00-component as an example and leave the rest as an exercise. Consider

µ = ν = 0 in (4.2.50):

g00(X) =∂Xα

∂τ

∂Xβ

∂τgαβ(X) . (4.2.51)

The only term that contributes to the l.h.s. is the one with α = β = 0. Consider for example

α = 0 and β = i. The off-diagonal component of the metric g0i is proportional to Bi, so it

is a first-order perturbation. But ∂Xi/∂τ is proportional to the first-order variable ξi, so the

product is second order and can be neglected. A similar argument holds for α = i and β = j.

Eq. (4.2.51) therefore reduces to

g00(X) =

(∂τ

∂τ

)2

g00(X) . (4.2.52)

Substituting (4.2.48) and (4.2.41), we get

a2(τ)(1 + 2A

)=(1 + T ′

)2a2(τ + T )

(1 + 2A

)=(1 + 2T ′ + · · ·

)(a(τ) + a′T + · · ·

)2(1 + 2A

)= a2(τ)

(1 + 2HT + 2T ′ + 2A+ · · ·

), (4.2.53)

where H ≡ a′/a is the Hubble parameter in conformal time. Hence, we find that at first order,

the metric perturbation A transforms as

A 7→ A = A− T ′ −HT . (4.2.54)

I leave it to you to repeat the argument for the other metric components and show that

Bi 7→ Bi = Bi + ∂iT − L′i , (4.2.55)

hij 7→ hij = hij − 2∂(iLj) − 2HTδij . (4.2.56)

Exercise.—Derive eqs. (4.2.55) and (4.2.56).

In terms of the SVT-decomposition, we get

A 7→ A− T ′ −HT , (4.2.57)

B 7→ B + T − L′ , Bi 7→ Bi − L′i , (4.2.58)

C 7→ C −HT − 1

3∇2L , (4.2.59)

E 7→ E − L , Ei 7→ Ei − Li , Eij 7→ Eij . (4.2.60)

Gauge-Invariant Perturbations

One way to avoid the gauge problems is to define special combinations of metric perturbations

that do not transform under a change of coordinates. These are the Bardeen variables:

Ψ ≡ A+H(B − E′) + (B − E′)′ , Φi ≡ E′i − Bi , Eij , (4.2.61)

Φ ≡ −C −H(B − E′) +1

3∇2E . (4.2.62)


Exercise.—Show that Ψ, Φ and Φi don’t change under a coordinate transformation.

These gauge-invariant variables can be considered as the ‘real’ spacetime perturbations since

they cannot be removed by a gauge transformation.

Gauge Fixing

An alternative (but related) solution to the gauge problem is to fix the gauge and keep track

of all perturbations (metric and matter). For example, we can use the freedom in the gauge

functions T and L in (4.2.48) to set two of the four scalar metric perturbations to zero:

• Newtonian gauge.—The choice

B = E = 0 , (4.2.63)

gives the metric

ds2 = a2(τ)[(1 + 2Ψ)dτ2 − (1− 2Φ)δijdx

idxj]. (4.2.64)

Here, we have renamed the remaining two metric perturbations, A ≡ Ψ and C ≡ −Φ, in

order to make contact with the Bardeen potentials in (4.2.61) and (4.2.62). For perturba-

tions that decay at spatial infinity, the Newtonian gauge is unique (i.e. the gauge is fixed

completely).2 In this gauge, the physics appears rather simple since the hypersurfaces of

constant time are orthogonal to the worldlines of observers at rest in the coordinates (since

B = 0) and the induced geometry of the constant-time hypersurfaces is isotropic (since

E = 0). In the absence of anisotropic stress, Ψ = Φ. Note the similarity of the metric to

the usual weak-field limit of GR about Minkowski space; we shall see that Ψ plays the role

of the gravitational potential. Newtonian gauge will be our preferred gauge for studying

the formation of large-scale structures (Chapter 5) and CMB anisotropies (Chapter ??).

• Spatially-flat qauge.—A convenient gauge for computing inflationary perturbations is

C = E = 0 . (4.2.65)

In this gauge, we will be able to focus most directly on the fluctuations in the inflaton

field δφ (see Chapter 6) .

4.2.2 Perturbed Matter

In Chapter 1, we showed that the matter in a homogeneous and isotropic universe has to take

the form of a perfect fluid

Tµν = (ρ+ P )UµUν − P δµν , (4.2.66)

where Uµ = aδ0µ, Uµ = a−1δµ0 for a comoving observer. Now, we consider small perturbations of

the stress-energy tensor

Tµν = Tµν + δTµν . (4.2.67)

2More generally, a gauge transformation that corresponds to a small, time-dependent but spatially constant

boost – i.e. Li(τ) and a compensating time translation with ∂iT = Li(τ) to keep the constant-time hypersurfaces

orthogonal – will preserve Eij = 0 and Bi = 0 and hence the form of the metric in eq. (4.4.168). However, such

a transformation would not preserve the decay of the perturbations at infinity.


Perturbations of the Stress-Energy Tensor

In a perturbed universe, the energy density ρ, the pressure P and the four-velocity Uµ can

be functions of position. Moreover, the stress-energy tensor can now have a contribution from

anisotropic stress, Πµν . The perturbation of the stress-energy tensor is

δTµν = (δρ+ δP )UµUν + (ρ+ P )(δUµUν + UµδUν)− δP δµν −Πµν . (4.2.68)

The spatial part of the anisotropic stress tensor can be chosen to be traceless, Πii = 0, since its

trace can always be absorbed into a redefinition of the isotropic pressure, P . The anisotropic

stress tensor can also be chosen to be orthogonal to Uµ, i.e. UµΠµν = 0. Without loss of

generality, we can then set Π00 = Π0

i = 0. In practice, the anisotropic stress will always be

negligible in these lectures. We will keep it for now, but at some point we will drop it.

Perturbations in the four-velocity can induce non-vanishing energy flux, T 0j , and momen-

tum density, T i0. To find these, let us compute the perturbed four-velocity in the perturbed

metric (4.2.41). Since gµνUµUν = 1 and gµνU

µUν = 1, we have, at linear order,

δgµνUµUν + 2UµδU

µ = 0 . (4.2.69)

Using Uµ = a−1δ0µ and δg00 = 2a2A, we find δU0 = −Aa−1. We then write δU i ≡ vi/a, where

vi ≡ dxi/dτ is the coordinate velocity, so that

Uµ = a−1[1−A, vi] . (4.2.70)

From this, we derive

U0 = g00U0 +

O(2)︷︸︸︷g0iU

i = a2(1 + 2A)a−1(1−A) = a(1 +A) , (4.2.71)

Ui = gi0U0 + gijU

j = −a2Bia−1 − a2δija

−1vj = −a(Bi + vi) , (4.2.72)

i.e.

Uµ = a[1 +A,−(vi +Bi)] . (4.2.73)

Using (4.2.70) and (4.2.73) in (4.2.68), we find

δT 00 = δρ , (4.2.74)

δT i0 = (ρ+ P )vi , (4.2.75)

δT 0j = −(ρ+ P )(vj +Bj) , (4.2.76)

δT ij = −δPδij −Πij . (4.2.77)

We will use qi for the momentum density (ρ + P )vi. If there are several contributions to the

stress-energy tensor (e.g. photons, baryons, dark matter, etc.), they are added: Tµν =∑

I TIµν .

This implies

δρ =∑I

δρI , δP =∑I

δPI , qi =∑I

qiI , Πij =∑I

ΠijI . (4.2.78)

We see that the perturbations in the density, pressure and anisotropic stress simply add. The

velocities do not add, but the momentum densities do.


Finally, we note that the SVT decomposition can also be applied to the perturbations of the

stress-energy tensor: δρ and δP have scalar parts only, qi has scalar and vector parts,

qi = ∂iq + qi , (4.2.79)

and Πij has scalar, vector and tensor parts,

Πij = ∂〈i∂j〉Π + ∂(iΠj) + Πij . (4.2.80)

Gauge Transformations

Under the coordinate transformation (4.2.48), the stress-energy tensor transform as

Tµν(X) =∂Xµ

∂Xα

∂Xβ

∂XνTαβ(X) . (4.2.81)

Evaluating this for the different components, we find

δρ 7→ δρ− T ρ ′ , (4.2.82)

δP 7→ δP − T P ′ , (4.2.83)

qi 7→ qi + (ρ+ P )L′i , (4.2.84)

vi 7→ vi + L′i , (4.2.85)

Πij 7→ Πij . (4.2.86)

Exercise.—Confirm eqs. (4.2.82)–(4.2.86). [Hint: First, convince yourself that the inverse of a matrix

of the form 1 + ε, were 1 is the identity and ε is a small perturbation, is 1− ε to first order in ε.]

Gauge-Invariant Perturbations

There are various gauge-invariant quantities that can be formed from metric and matter vari-

ables. One useful combination is

ρ∆ ≡ δρ+ ρ ′(v +B) , (4.2.87)

where vi = ∂iv. The quantity ∆ is called the comoving-gauge density perturbation.

Exercise.—Show that ∆ is gauge-invariant.

Gauge Fixing

Above we used our gauge freedom to set two of the metric perturbations to zero. Alternatively,

we can define the gauge in the matter sector:

• Uniform density gauge.—We can use the freedom in the time-slicing to set the total density

perturbation to zero

δρ = 0 . (4.2.88)


• Comoving gauge.—Similarly, we can ask for the scalar momentum density to vanish,

q = 0 . (4.2.89)

Fluctuations in comoving gauge are most naturally connected to the inflationary initial

conditions. This will be explained in §4.3.1 and Chapter 6.

There are different versions of uniform density and comoving gauge depending on which of the

metric fluctuations is set to zero. In these lectures, we will choose B = 0.

Adiabatic Fluctuations

Simple inflation models predict initial fluctuations that are adiabatic (see Chapter 6). Adiabatic

perturbations have the property that the local state of matter (determined, for example, by the

energy density ρ and the pressure P ) at some spacetime point (τ,x) of the perturbed universe

is the same as in the background universe at some slightly different time τ + δτ(x). (Notice that

the time shift varies with location x!) We can thus view adiabatic perturbations as some parts

of the universe being “ahead” and others “behind” in the evolution. If the universe is filled with

multiple fluids, adiabatic perturbations correspond to perturbations induced by a common, local

shift in time of all background quantities; e.g. adiabatic density perturbations are defined as

δρI(τ,x) ≡ ρI(τ + δτ(x))− ρI(τ) = ρ ′I δτ(x) , (4.2.90)

where δτ is the same for all species I. This implies

δτ =δρIρ ′I

=δρJρ ′J

for all species I and J . (4.2.91)

Using3 ρ ′I = −3H(1 + wI)ρI , we can write this as

δI1 + wI

=δJ

1 + wJfor all species I and J , (4.2.92)

where we have defined the fractional density contrast

δI ≡δρIρI

. (4.2.93)

Thus, for adiabatic perturbations, all matter components (wm ≈ 0) have the same fractional

perturbation, while all radiation perturbations (wr = 13) obey

δr =4

3δm . (4.2.94)

It follows that for adiabatic fluctuations, the total density perturbation,

δρtot = ρtotδtot =∑I

ρIδI , (4.2.95)

is dominated by the species that is dominant in the background since all the δI are comparable.

We will have more to say about adiabatic initial conditions in §4.3.

3If there is no energy transfer between the fluid components at the background level, the energy continuity

equation is satisfied by them separately.


Isocurvature Fluctuations

The complement of adiabatic perturbations are isocurvature perturbations. While adiabatic

perturbations correspond to a change in the total energy density, isocurvature perturbations

only correspond to perturbations between the different components. Eq. (4.2.92) suggests the

following definition of isocurvature fluctuations

SIJ ≡δI

1 + wI− δJ

1 + wJ. (4.2.96)

Single-field inflation predicts that the primordial perturbations are purely adiabatic, i.e. SIJ =

0, for all species I and J . Moreover, all present observational data is consistent with this

expectation. We therefore won’t consider isocurvature fluctuations further in these lectures.

4.2.3 Linearised Evolution Equations

Our next task is to derive the perturbed Einstein equations, δGµν = 8πGδTµν , from the per-

turbed metric and the perturbed stress-energy tensor. We will work in Newtonian gauge with

gµν = a2

(1 + 2Ψ 0

0 −(1− 2Φ)δij

). (4.2.97)

In these lectures, we will never encounter situations where anisotropic stress plays a significant

role. From now on, we will therefore set anisotropic stress to zero, Πij = 0. As we will see, this

enforces Φ = Ψ.

Perturbed Connection Coefficients

To derive the field equations, we first require the perturbed connection coefficients. Recall that

Γµνρ =1

2gµλ (∂νgλρ + ∂ρgλν − ∂λgνρ) . (4.2.98)

Since the metric (4.2.97) is diagonal, it is simple to invert

gµν =1

a2

(1− 2Ψ 0

0 −(1 + 2Φ)δij

). (4.2.99)

Substituting (4.2.97) and (4.2.99) into (4.2.98), gives

Γ000 = H+ Ψ′ , (4.2.100)

Γ00i = ∂iΨ , (4.2.101)

Γi00 = δij∂jΨ , (4.2.102)

Γ0ij = Hδij −

[Φ′ + 2H(Φ + Ψ)

]δij , (4.2.103)

Γij0 = Hδij − Φ′δij , (4.2.104)

Γijk = −2δi(j∂k)Φ + δjkδil∂lΦ . (4.2.105)

I will work out Γ000 as an example and leave the remaining terms as an exercise.


Example.—From the definition of the Christoffel symbol we have

Γ000 =

1

2g00(2∂0g00 − ∂0g00)

=1

2g00∂0g00 . (4.2.106)

Substituting the metric components, we find

Γ000 =

1

2a2(1− 2Ψ)∂0[a2(1 + 2Ψ)]

= H+ Ψ′ , (4.2.107)

at linear order in Ψ.

Exercise.—Derive eqs. (4.2.101)–(4.2.105).

Perturbed Stress-Energy Conservation

Equipped with the perturbed connection, we can immediately derive the perturbed conservation

equations from

∇µTµν = 0

= ∂µTµν + ΓµµαT

αν − ΓαµνT

µα . (4.2.108)

Continuity Equation

Consider first the ν = 0 component

∂0T0

0 + ∂iTi0 + Γµµ0T

00 + ΓµµiT

i0︸︷︷︸

O(2)

−Γ000T

00 − Γ0

i0Ti0︸︷︷︸

O(2)

−Γi00T0i︸︷︷︸

O(2)

−Γij0Tji = 0 . (4.2.109)

Substituting the perturbed stress-energy tensor and the connection coefficients gives

∂0(ρ+ δρ) + ∂iqi + (H+ Ψ′ + 3H− 3Φ′)(ρ+ δρ)

− (H+ Ψ′)(ρ+ δρ)− (H− Φ′)δij[− (P + δP )δji

]= 0 , (4.2.110)

and hence

ρ ′ + δρ′ + ∂iqi + 3H(ρ+ δρ)− 3ρΦ′ + 3H(P + δP )− 3P Φ′ = 0 . (4.2.111)

Writing the zeroth-order and first-order parts separately, we get

ρ ′ = −3H(ρ+ P ) , (4.2.112)

δρ′ = −3H(δρ+ δP ) + 3Φ′(ρ+ P )−∇ · q . (4.2.113)

The zeroth-order part (4.2.112) simply is the conservation of energy in the homogeneous back-

ground. Eq. (4.2.113) describes the evolution of the density perturbation. The first term on

the right-hand side is just the dilution due to the background expansion (as in the background


equation), the ∇ · q term accounts for the local fluid flow due to peculiar velocity, and the Φ′

term is a purely relativistic effect corresponding to the density changes caused by perturbations

to the local expansion rate [(1−Φ)a is the “local scale factor” in the spatial part of the metric

in Newtonian gauge].

It is convenient to write the equation in terms of the fractional overdensity and the 3-velocity,

δ ≡ δρ

ρand v =

q

ρ+ P. (4.2.114)

Eq. (4.2.113) then becomes

δ′ +

(1 +

P

ρ

)(∇ · v − 3Φ′

)+ 3H

(δP

δρ− P

ρ

)δ = 0 . (4.2.115)

This is the relativistic version of the continuity equation. In the limit P ρ, we recover the

Newtonian continuity equation in conformal time, δ′ + ∇ · v − 3Φ′ = 0, but with a general-

relativistic correction due to the perturbation to the rate of exansion of space. This correction

is small on sub-horizon scales (k H) — we will prove this rigorously in Chapter 5.

Euler Equation

Next, consider the ν = i component of eq. (4.2.108),

∂µTµi + ΓµµρT

ρi − ΓρµiT

µρ = 0 , (4.2.116)

and hence

∂0T0i + ∂jT

ji + Γµµ0T

0i + ΓµµjT

ji − Γ0

0iT0

0 − Γ0jiT

j0 − Γj0iT

0j − ΓjkiT

kj = 0 . (4.2.117)

Using eqs. (4.2.74)–(4.2.77), with T 0i = −qi in Newtonian gauge, eq. (4.2.117) becomes

−q′i + ∂j

[−(P + δP )δji

]− 4Hqi − (∂jΨ− 3∂jΦ)P δji − ∂iΨρ

−Hδjiqj +Hδji qj +(−2δj(i∂k)Φ + δkiδ

jl∂lΦ)P δkj︸︷︷︸

−3∂iΦ P

= 0 , (4.2.118)

or

−q′i − ∂iδP − 4Hqi − (ρ+ P )∂iΨ = 0 . (4.2.119)

Using eqs. (4.2.112) and (4.2.114), we get

v′ +Hv − 3H P′

ρ ′v = −∇δP

ρ+ P−∇Ψ . (4.2.120)

This is the relativistic version of the Euler equation for a viscous fluid. Pressure gradients

(∇δP ) and gravitational infall (∇Ψ) drive v′. The equation captures the redshifting of peculiar

velocities (Hv) and includes a small correction for relativistic fluids (P ′/ρ ′). Adiabatic fluctua-

tions satisfy P ′/ρ ′ = c2s. Non-relativistic matter fluctuations have a very small sound speed, so

the relativistic correction in the Euler equation (4.2.120) is much smaller than the redshifting


term. The limit P ρ then reproduces the Euler equation (4.1.25) of the linearised Newtonian

treatment.

Eqs. (4.2.115) and (4.2.120) apply for the total matter and velocity, and also separately for any

non-interacting components so that the individual stress-energy tensors are separately conserved.

Once an equation of state of the matter (and other constitutive relations) are specified, we just

need the gravitational potentials Ψ and Φ to close the system of equations. Equations for Ψ and

Φ follow from the perturbed Einstein equations.

Perturbed Einstein Equations

Let us now compute the linearised Einstein equation in Newtonian gauge. We require the

perturbation to the Einstein tensor, Gµν ≡ Rµν − 12Rgµν , so we first need to calculate the

perturbed Ricci tensor Rµν and scalar R.

Ricci tensor.—We recall that the Ricci tensor can be expressed in terms of the connection as

Rµν = ∂λΓλµν − ∂νΓλµλ + ΓλλρΓρµν − ΓρµλΓλνρ . (4.2.121)

Substituting the perturbed connection coefficients (4.2.100)–(4.2.105), we find

R00 = −3H′ +∇2Ψ + 3H(Φ′ + Ψ′) + 3Φ′′ , (4.2.122)

R0i = 2∂iΦ′ + 2H∂iΨ , (4.2.123)

Rij =[H′ + 2H2 − Φ′′ +∇2Φ− 2(H′ + 2H2)(Φ + Ψ)−HΨ′ − 5HΦ′

]δij (4.2.124)

+ ∂i∂j(Φ−Ψ) .

I will derive R00 here and leave the others as an exercise.

Example.—The 00 component of the Ricci tensor is

R00 = ∂ρΓρ00 − ∂0Γρ0ρ + Γα00Γραρ − Γα0ρΓ

ρ0α . (4.2.125)

When we sum over ρ, the terms with ρ = 0 cancel so we need only consider summing over ρ = 1, 2, 3,

i.e.

R00 = ∂iΓi00 − ∂0Γi0i + Γα00Γiαi − Γα0iΓ

i0α

= ∂iΓi00 − ∂0Γi0i + Γ0

00Γi0i + Γj00Γiji︸︷︷︸O(2)

−Γ00iΓ

i00︸︷︷︸

O(2)

−Γj0iΓi0j

= ∇2Ψ− 3∂0(H− Φ′) + 3(H+ Ψ′)(H− Φ′)− (H− Φ′)2δji δij

= −3H′ +∇2Ψ + 3H(Φ′ + Ψ′) + 3Φ′′ . (4.2.126)

Exercise.—Derive eqs. (4.2.123) and (4.2.124).

Ricci scalar.—It is now relatively straightforward to compute the Ricci scalar

R = g00R00 + 2 g0iR0i︸︷︷︸0

+gijRij . (4.2.127)


It follows that

a2R = (1− 2Ψ)R00 − (1 + 2Φ)δijRij

= (1− 2Ψ)[−3H′ +∇2Ψ + 3H(Φ′ + Ψ′) + 3Φ′′

]− 3(1 + 2Φ)

[H′ + 2H2 − Φ′′ +∇2Φ− 2(H′ + 2H2)(Φ + Ψ)−HΨ′ − 5HΦ′

]− (1 + 2Φ)∇2(Φ−Ψ) . (4.2.128)

Dropping non-linear terms, we find

a2R = −6(H′ +H2) + 2∇2Ψ− 4∇2Φ + 12(H′ +H2)Ψ + 6Φ′′ + 6H(Ψ′ + 3Φ′) . (4.2.129)

Einstein tensor.—Computing the Einstein tensor is now just a matter of collecting our previous

results. The 00 component is

G00 = R00 −1

2g00R

= −3H′ +∇2Ψ + 3H(Φ′ + Ψ′) + 3Φ′′ + 3(1 + 2Ψ)(H′ +H2)

− 1

2

[2∇2Ψ− 4∇2Φ + 12(H′ +H2)Ψ + 6Φ′′ + 6H(Ψ′ + 3Φ′)

]. (4.2.130)

Most of the terms cancel leaving the simple result

G00 = 3H2 + 2∇2Φ− 6HΦ′ . (4.2.131)

The 0i component of the Einstein tensor is simply R0i since g0i = 0 in Newtonian gauge:

G0i = 2∂i(Φ′ +HΨ) . (4.2.132)

The remaining components are

Gij = Rij −1

2gijR

=[H′ + 2H2 − Φ′′ +∇2Φ− 2(H′ + 2H2)(Φ + Ψ)−HΨ′ − 5HΦ′

]δij + ∂i∂j(Φ−Ψ)

− 3(1− 2Φ)(H′ +H2)δij

+1

2

[2∇2Ψ− 4∇2Φ + 12(H′ +H2)Ψ + 6Φ′′ + 6H(Ψ′ + 3Φ′)

]δij . (4.2.133)

This neatens up (only a little!) to give

Gij = −(2H′ +H2)δij +[∇2(Ψ− Φ) + 2Φ′′ + 2(2H′ +H2)(Φ + Ψ) + 2HΨ′ + 4HΦ′

]δij

+ ∂i∂j(Φ−Ψ) . (4.2.134)

Einstein Equations

Substituting the perturbed Einstein tensor, metric and stress-energy tensor into the Einstein

equation gives the equations of motion for the metric perturbations and the zeroth-order Fried-

mann equations:

• Let us start with the trace-free part of the ij equation, Gij = 8πGTij . Since we have

dropped anisotropic stress there is no source on the right-hand side. From eq. (4.2.134),

we get

∂〈i∂j〉(Φ−Ψ) = 0 . (4.2.135)


Had we kept anisotropic stress, the right-hand side would be −8πGa2Πij . In the absence

of anisotropic stress4 (and assuming appropriate decay at infinity), we get5

Φ = Ψ . (4.2.136)

There is then only one gauge-invariant degree of freedom in the metric. In the following,

we will write all equations in terms of Φ.

• Next, we consider the 00 equation, G00 = 8πGT00. Using eq. (4.2.131), we get

3H2 + 2∇2Φ− 6HΦ′ = 8πGg0µTµ

0

= 8πG(g00T

00 + g0iT

i0

)= 8πGa2(1 + 2Φ)(ρ+ δρ)

= 8πGa2ρ(1 + 2Φ + δ) . (4.2.137)

The zeroth-order part gives

H2 =8πG

3a2ρ , (4.2.138)

which is just the Friedmann equation. The first-order part of eq. (4.2.137) gives

∇2Φ = 4πGa2ρδ + 8πGa2ρΦ + 3HΦ′ . (4.2.139)

which, on using eq. (4.2.138), reduces to

∇2Φ = 4πGa2ρδ + 3H(Φ′ +HΦ) . (4.2.140)

• Moving on to 0i equation, G0i = 8πGT0i, with

T0i = g0µTµi = g00T

0i = g00T

0i = −a2qi . (4.2.141)

It follows that

∂i(Φ′ +HΦ) = −4πGa2qi . (4.2.142)

If we write qi = (ρ+P )∂iv and assume the perturbations decay at infinity, we can integrate

eq. (4.2.142) to get

Φ′ +HΦ = −4πGa2(ρ+ P )v . (4.2.143)

• Substituting eq. (4.2.143) into the 00 Einstein equation (4.2.140) gives

∇2Φ = 4πGa2ρ∆ , where ρ∆ ≡ ρδ − 3H(ρ+ P )v . (4.2.144)

4In reality, neutrinos develop anisotropic stress after neutrino decoupling (i.e. they do not behave like a perfect

fluid). Therefore, Φ and Ψ actually differ from each other by about 10% in the time between neutrino decoupling

and matter-radiation equality. After the universe becomes matter-dominated, the neutrinos become unimportant,

and Φ and Ψ rapidly approach each other. The same thing happens to photons after photon decoupling, but the

universe is then already matter-dominated, so they do not cause a significant Φ−Ψ difference.5In Fourier space, eq. (4.2.135) becomes(

kikj − 13δijk

2) (Φ−Ψ) = 0 .

For finite k, we therefore must have Φ = Ψ. For k = 0, Φ−Ψ = const. would be a solution. However, the constant

must be zero, since the mean of the perturbations vanishes.


This is of the form of a Poisson equation, but with source density given by the gauge-

invariant variable ∆ of eq. (4.2.87) since B = 0 in the Newtonian gauge. Let us introduce

comoving hypersurfaces as those that are orthogonal to the worldlines of a set of observers

comoving with the total matter (i.e. they see qi = 0) and are the constant-time hypersur-

faces in the comoving gauge for which qi = 0 and Bi = 0. It follows that ∆ is the fractional

overdensity in the comoving gauge and we see from eq. (4.2.144) that this is the source

term for the gravitational potential Φ.

• Finally, we consider the trace-part of the ij equation, i.e. Gii = 8πGT ii. We compute the

left-hand side from eq. (4.2.134) (with Φ = Ψ),

Gii = giµGµi

= gikGki

= −a−2(1 + 2Φ)δik[−(2H′ +H2)δki +

(2Φ′′ + 6HΦ′ + 4(2H′ +H2)Φ

)δki]

= −3a−2[−(2H′ +H2) + 2

(Φ′′ + 3HΦ′ + (2H′ +H2)Φ

)]. (4.2.145)

We combine this with T ii = −3(P + δP ). At zeroth order, we find

2H′ +H2 = −8πGa2P , (4.2.146)

which is just the second Friedmann equation. At first order, we get

Φ′′ + 3HΦ′ + (2H′ +H2)Φ = 4πGa2δP . (4.2.147)

Of course, the Einstein equations and the energy and momentum conservation equations form

a redundant (but consistent!) set of equations because of the Bianchi identity. We can use

whichever subsets are most convenient for the particular problem at hand.

4.3 Conserved Curvature Perturbation

There is an important quantity that is conserved on super-Hubble scales for adiabatic fluctuations

irrespective of the equation of state of the matter: the comoving curvature perturbation. As we

will see below, the comoving curvature perturbation provides the essential link between the

fluctuations that we observe in the late-time universe (Chapter 5) and the primordial seed

fluctuations created by inflation (Chapter 6).

4.3.1 Comoving Curvature Perturbation

In some arbitrary gauge, let us work out the intrinsic curvature of surfaces of constant time.

The induced metric, γij , on these surfaces is just the spatial part of eq. (4.2.41), i.e.

γij ≡ a2 [(1 + 2C)δij + 2Eij ] . (4.3.148)

where Eij ≡ ∂〈i∂j〉E for scalar perturbations. In a tedious, but straightforward computation,

we derive the three-dimensional Ricci scalar associated with γij ,

a2R(3) = −4∇2

(C − 1

3∇2E

). (4.3.149)

In the following insert I show all the steps.


Derivation.—The connection corresponding to γij is

(3)Γijk =1

2γil (∂jγkl + ∂kγjl − ∂lγjk) , (4.3.150)

where γij is the inverse of the induced metric,

γij = a−2[(1− 2C)δij − 2Eij

]= a−2δij +O(1) . (4.3.151)

In order to compute the connection to first order, we actually only need the inverse metric to zeroth

order, since the spatial derivatives of the γij are all first order in the perturbations. We have

(3)Γijk = δil∂j (Cδkl + Ekl) + δil∂k (Cδjl + Ejl)− δil∂l (Cδjk + Ejk)

= 2δi(j∂k)C − δilδjk∂lC + 2∂(jEk)i − δil∂lEjk . (4.3.152)

The intrinsic curvature is the associated Ricci scalar, given by

R(3) = γik∂l(3)Γlik − γik∂k(3)Γlil + γik (3)Γlik

(3)Γmlm − γik (3)Γmil(3)Γlkm . (4.3.153)

To first order, this reduces to

a2R(3) = δik∂l(3)Γlik − δik∂k(3)Γlil . (4.3.154)

This involves two contractions of the connection. The first is

δik(3)Γlik = δik(

2δl(i∂k)C − δjlδik∂jC)

+ δik(

2∂(iEk)l − δjl∂jEik

)= 2δkl∂kC − 3δjl∂jC + 2∂iE

il − δjl∂j (δikEik)︸︷︷︸0

= −δkl∂kC + 2∂kEkl . (4.3.155)

The second is

(3)Γlil = δll∂iC + δli∂lC − ∂iC + ∂lEil + ∂iEl

l − ∂lEil

= 3∂iC . (4.3.156)

Eq. (4.3.154) therefore becomes

a2R(3) = ∂l(−δkl∂kC + 2∂kE

kl)− 3δik∂k∂iC

= −∇2C + 2∂i∂jEij − 3∇2C

= −4∇2C + 2∂i∂jEij . (4.3.157)

Note that this vanishes for vector and tensor perturbations (as do all perturbed scalars) since then

C = 0 and ∂i∂jEij = 0. For scalar perturbations, Eij = ∂〈i∂j〉E so

∂i∂jEij = δilδjm∂i∂j

(∂l∂mE −

1

3δlm∇2E

)= ∇2∇2E − 1

3∇2∇2E

=2

3∇4E . (4.3.158)

Finally, we get eq. (4.3.149).

We define the curvature perturbation as C − 13∇

2E. The comoving curvature perturbation R


is the curvature perturbation evaluated in the comoving gauge (Bi = 0 = qi). It will prove

convenient to have a gauge-invariant expression for R, so that we can evaluate it from the

perturbations in any gauge (for example, in Newtonian gauge). Since B and v vanish in the

comoving gauge, we can always add linear combinations of these to C − 13∇

2E to form a gauge-

invariant combination that equals R. Using eqs. (4.2.58)–(4.2.60) and (4.2.85), we see that the

correct gauge-invariant expression for the comoving curvature perturbation is

R = C − 1

3∇2E +H(B + v) . (4.3.159)

Exercise.—Show that R is gauge-invariant.

4.3.2 A Conservation Law

We now want to prove that the comoving curvature perturbation R is indeed conserved on large

scales and for adiabatic perturbations. We shall do so by working in the Newtonian gauge, in

which case

R = −Φ +Hv , (4.3.160)

since B = E = 0 and C ≡ −Φ. We can use the 0i Einstein equation (4.2.143) to eliminate the

peculiar velocity in favour of the gravitational potential and its time derivative:

R = −Φ− H(Φ′ +HΦ)

4πGa2(ρ+ P ). (4.3.161)

Taking a time derivative of (4.3.161) and using the evolution equations of the previous section,

we find

−4πGa2(ρ+ P )R′ = 4πGa2HδPnad +H P′

ρ ′∇2Φ , (4.3.162)

where we have defined the non-adiabatic pressure perturbation

δPnad ≡ δP −P ′

ρ ′δρ . (4.3.163)

Derivation.∗—We differentiate eq. (4.3.161) to find

−4πGa2(ρ+ P )R′ = 4πGa2(ρ+ P )Φ′ +H′(Φ′ +HΦ) +H(Φ′′ +H′Φ +HΦ′)

+H2(Φ′ +HΦ) + 3H2 P′

ρ ′(Φ′ +HΦ) , (4.3.164)

where we used ρ ′ = −3H(ρ+ P ). This needs to be cleaned up a bit. In the first term on the right,

we use the Friedmann equation to write 4πGa2(ρ + P ) as H2 − H′. In the last term, we use the

Poisson equation (4.2.140) to write 3H(Φ′ +HΦ) as (∇2Φ− 4πGa2ρδ). We then find

−4πGa2(ρ+ P )R′ = (H2 −H′)Φ′ +H′(Φ′ +HΦ) +H(Φ′′ +H′Φ +HΦ′)

+H2(Φ′ +HΦ) +H P′

ρ ′(∇2Φ− 4πGa2ρδ

). (4.3.165)


Adding and subtracting 4πGa2HδP on the right-hand side and simplifying gives

−4πGa2(ρ+ P )R′ = H[Φ′′ + 3HΦ′ + (2H′ +H2)Φ− 4πGa2δP

]+ 4πGa2HδPnad +H P

′

ρ ′∇2Φ , (4.3.166)

where δPnad was defined in (4.3.163). The first term on the right-hand side vanishes by eq. (4.2.147),

so we obtain eq. (4.3.162).

Exercise.—Show that δPnad is gauge-invariant.

The non-adiabatic pressure δPnad vanishes for a barotropic equation of state, P = P (ρ) (and,

more generally, for adiabatic fluctuations in a mixture of barotropic fluids). In that case, the

right-hand side of eq. (4.3.162) scales as Hk2Φ ∼ Hk2R, so that

d lnRd ln a

∼(k

H

)2

. (4.3.167)

Hence, we find that R doesn’t evolve on super-Hubble scales, k H. This means that the value

of R that we will compute at horizon crossing during inflation (Chapter 6) survives unaltered

until later times.

4.4 Summary

We have derived the linearised evolution equations for scalar perturbations in Newtonian gauge,

where the metric has the following form

ds2 = a2(τ)[(1 + 2Ψ)dτ2 − (1− 2Φ)δijdx

idxj]. (4.4.168)

In these lectures, we won’t encounter situations where anisotropic stress plays a significant role,

so we will always be able to set Ψ = Φ.

• The Einstein equations then are

∇2Φ− 3H(Φ′ +HΦ) = 4πGa2 δρ , (4.4.169)

Φ′ +HΦ = −4πGa2(ρ+ P )v , (4.4.170)

Φ′′ + 3HΦ′ + (2H′ +H2)Φ = 4πGa2 δP . (4.4.171)

The source terms on the right-hand side should be interpreted as the sum over all relevant

matter components (e.g. photons, dark matter, baryons, etc.). The Poisson equation takes

a particularly simple form if we introduce the comoving gauge density contrast

∇2Φ = 4πGa2ρ∆ . (4.4.172)

• From the conservation of the stress-tensor, we derived the relativistic generalisations of

the continuity equation and the Euler equation

δ′ + 3H(δP

δρ− P

ρ

)δ = −

(1 +

P

ρ

)(∇ · v − 3Φ′

), (4.4.173)

v′ + 3H(

1

3− P ′

ρ ′

)v = −∇δP

ρ+ P−∇Φ . (4.4.174)


These equations apply for the total matter and velocity, and also separately for any non-

interacting components so that the individual stress-energy tensors are separately con-

served.

• A very important quantity is the comoving curvature perturbation

R = −Φ− H(Φ′ +HΦ)

4πGa2(ρ+ P ). (4.4.175)

We have shown thatR doesn’t evolve on super-Hubble scales, k H, unless non-adiabatic

pressure is significant. This fact is crucial for relating late-time observables, such as the

distributions of galaxies (Chapter 5), to the initial conditions from inflation (Chapter 6).

5 Structure Formation

In the previous chapter, we derived the evolution equations for all matter and metric perturba-

tions. In principle, we could now solve these equations. The complex interactions between the

different species (see fig. 5.1) means that we get a large number of coupled differential equations.

This set of equations is easy to solve numerically and this is what is usually done. However, our

goal in this chapter is to obtain some analytical insights into the basic qualitative features of

the solutions.

Metric

DarkEnergy

Electrons

Photons

Neutrions

DarkMatter

Protons

Radiatio

n

Baryons Matter

ThomsonScattering

CoulombScattering

Figure 5.1: Interactions between the different forms of matter in the universe.

5.1 Initial Conditions

Any mode of interest for observations today was outside the Hubble radius if we go back suffi-

ciently far into the past. Inflation sets the initial condition for these superhorizon modes. The

prediction from inflation (see Ch. 6) is presented most conveniently in terms of a spectrum of

fluctuations for the curvature perturbation R. Eq. (4.4.175) relates this to the gravitational

potential Φ in Newtonian gauge

R = −Φ− 2

3(1 + w)

(Φ′

H+ Φ

), (5.1.1)

where w is the equation of state of the background. For adiabatic perturbations, we have

c2s ≈ w and a combination of Einstein equations imply a closed form evolution equation for the

gravitational potential

Φ ′′ + 3(1 + w)HΦ ′ + wk2Φ = 0 . (5.1.2)

101

102 5. Structure Formation

Notice that in deriving (5.1.2) we have assumed a constant equation of state. It therefore only

applies if a single component dominates the universe. For the more general case, you should

consult (4.4.171).

Exercise.—Derive eq. (5.1.2) from the Einstein equations.

5.1.1 Superhorizon Limit

On superhorizon scales, k H, we can drop the last term in (5.1.2). The growing-mode solution

then is

Φ = const. (superhorizon) . (5.1.3)

Notice that this superhorizon solution is independent of the equation of state w (as long as

w = const.). In particular, the gravitational potential is frozen outside the horizon during both

the radiation and matter eras.

The Poisson equation (4.4.169) relates the gravitational potential to the total Newtonian-gauge

density contrast

δ = −2

3

k2

H2Φ− 2

HΦ ′ − 2Φ , (5.1.4)

where we have used 32H

2 = 4πGa2ρ. On superhorizon scales, only the decaying mode contributes

to Φ′. The first and second term in (5.1.4) then are of the same order and both are much smaller

than the third term. We therefore get

δ ≈ −2Φ = const. , (5.1.5)

so δ is also frozen on superhorizon scales. For adiabatic initial conditions, we can relate the

primordial potential Φ to the fluctuations in both the matter and the radiation:

δm =3

4δr ≈ −

3

2ΦRD , (5.1.6)

where we have used that δr ≈ δ for adiabatic perturbations during the radiation era. On

superhorizon scales, the density perturbations are therefore simply proportional to the curvature

perturbation set up by inflation.

5.1.2 Radiation-to-Matter Transition

We have seen that the gravitational potential is frozen on superhorizon scales as long as the

equation of state of the background is constant. However, unlike the curvature perturbation R,

the gravitational doesn’t stay constant when the equation of state changes. To follow the

evolution of Φ through the radiation-to-matter transition, we exploit the conservation of R.

In the superhorizon limit, the comoving curvature perturbation (4.4.175) becomes

R = −5 + 3w

3 + 3wΦ (superhorizon) . (5.1.7)

This provides an important link between the source term for the evolution of fluctuations (Φ)

and the primordial initial conditions set up by inflation (R). Evaluating (5.1.7) for w = 13 and


w = 0 relates the amplitudes of Φ during the radiation era and the matter eta

R = −3

2ΦRD = −5

3ΦMD ⇒ ΦMD =

9

10ΦRD , (5.1.8)

where we have used that R = const. throughout. We see that the gravitational potential

decreases by a factor of 9/10 in the transition from radiation-dominated to matter-dominated.

5.2 Evolution of Fluctuations

We wish to understand what happens to the superhorizon initial conditions, when modes enter

the horizon. We will first study the evolution of the gravitational potential (§5.2.1), and then

the perturbations in radiation (§5.2.2), matter (§5.2.3) and baryons (§5.2.4).

5.2.1 Gravitational Potential

To determine the evolution of Φ during both the radiation era and the matter era, we simply

have to specialise (5.1.2) to w = 13 and w = 0, respectively.

Radiation Era

In the radiation era, w = 13 , we get

Φ ′′ +4

τΦ ′ +

k2

3Φ = 0 . (5.2.9)

This equation has the following exact solution

Φk(τ) = Akj1(x)

x+Bk

n1(x)

x, x ≡ 1√

3kτ , (5.2.10)

where the subscript k indicates that the solution can have different amplitudes for each value

of k. The size of the initial fluctuations as a function of wavenumber will be a prediction of

inflation. The functions j1(x) and n1(x) in (5.2.10) are the spherical Bessel and Neumann

functions

j1(x) =sinx

x2− cosx

x=x

3+O(x3) , (5.2.11)

n1(x) = −cosx

x2− sinx

x= − 1

x2+O(x0) . (5.2.12)

Since n1(x) blows up for small x (early times), we reject that solution on the basis of initial

conditions, i.e. we set Bk ≡ 0. We match the constant Ak to the primordial value of the

potential, Φk(0) = −23Rk(0). Using (5.2.11), we find

Φk(τ) = −2Rk(0)

(sinx− x cosx

x3

)(all scales) . (5.2.13)

Notice that (5.2.13) is valid on all scales. Outside the (sound) horizon, x = 1√3kτ 1, the

solution approaches Φ = const., while on subhorizon scales, x 1, we get

Φk(τ) ≈ −6Rk(0)cos(

1√3kτ)

(kτ)2(subhorizon) . (5.2.14)

During the radiation era, subhorizon modes of Φ therefore oscillate with frequency 1√3k and an

amplitude that decays as τ−2 ∝ a−2 (see fig. 5.2). Remember this.


Matter Era

In the matter era, w = 0, the evolution of the potential is

Φ′′ +6

τΦ′ = 0 , (5.2.15)

whose solution is

Φ ∝

const.

τ−5 ∝ a−5/2. (5.2.16)

We conclude that the gravitational potential is frozen on all scales during matter domination.

Summary

Fig. 5.2 shows the evolution of the gravitational potential for difference wavelengths. As pre-

dicted, the potential is constant when the modes are outside the horizon. Two of the modes enter

the horizon during the radiation era. While they are inside the horizon during the radiation

era their amplitudes decrease as a−2. The resulting amplitudes in the matter era are therefore

strongly suppressed. During the matter era the potential is constant on all scales. The longest

wavelength mode in the figure enters the horizon during the matter era, so its amplitude is only

suppressed by the factor of 910 coming from the radiation-to-matter transition.

Figure 5.2: Numerical solutions for the linear evolution of the gravitational potential.

5.2.2 Radiation

In this section, we wish to determine the evolution of perturbations in the radiation density.

Radiation Era

In the radiation era, perturbations in the radiation density dominate (for adiabatic initial con-

ditions). Given the solution (5.2.13) for Φ during the radiation era, we therefore immediately


obtain a solution for the density contrast of radiation (δr or ∆r) via the Poisson equation

δr = −2

3(kτ)2Φ− 2τΦ′ − 2Φ , (5.2.17)

∆r = −2

3(kτ)2Φ . (5.2.18)

We see that while δr is constant outside the horizon, ∆r grows as τ2 ∝ a2. Inside the horizon,1

δr ≈ ∆r = −2

3(kτ)2Φ = 4R(0) cos

(1√3kτ

), (5.2.19)

which is the solution to

δ′′r −1

3∇2δr = 0 . (5.2.20)

We see that subhorizon fluctuations in the radiation density oscillate with constant amplitude

around δr = 0.

Matter Era

In the matter era, radiation perturbations are subdominant. Their evolution then has to be

determined from the conservation equations. On subhorizon scales, we have

(C) δ ′r = −4

3∇ · vr

(E) v′r = −1

4∇δr −∇Φ

δ′′r −1

3∇2δr =

4

3∇2Φ = const. (5.2.21)

This is the equation of motion of a harmonic oscillator with constant driving force. During the

matter era, the subhorizon fluctuations in the radiation density therefore oscillate with constant

amplitude around a shifted equilibrium point, δr = −4ΦMD(k). Here, ΦMD(k) is the k-dependent

amplitude of the gravitational potential in the matter era; cf. fig. 5.2.

Summary

The acoustic oscillations in the perturbed radiation density are what gives rise to the peaks in

the spectrum of CMB anisotropies (see fig. 6.5 in §6.6.2) This will be analysed in much more

detail in the Advanced Cosmology course next term.

5.2.3 Dark Matter

In this section, we are interested in the evolution of matter fluctuations from early times (during

the radiation era) until late times (when dark energy starts to dominate).

Early Times

At early times, the universe was dominated by a mixture of radiation (r) and pressureless

matter (m). For now, we ignore baryons (but see §5.2.4). The conformal Hubble parameter is

H2 =H2

0Ω2m

Ωr

(1

y+

1

y2

), y ≡ a

aeq. (5.2.22)

1We see that well inside the horizon, the density perturbations in the comoving and Newtonian gauge coincide.

This is indicative of the general result that there are no gauge ambiguities inside the horizon.


We wish to determine how matter fluctuations evolve on subhorizon scales from the radiation

era until the matter era. We consider the evolution equations for the matter density contrast

and velocity:

(C) δ ′m = −∇ · vm

(E) v′m = −Hvm −∇Φ

δ′′m +Hδ′m = ∇2Φ . (5.2.23)

In general, the potential Φ is sourced by the total density fluctuation. However, we have seen

that perturbations in the radiation density oscillate rapidly on small scales. The time-averaged

gravitational potential is therefore only sourced by the matter fluctuations, and the fluctuations

in the radiation can be neglected (see Weinberg, astro-ph/0207375 for further discussion). The

evolution of the matter perturbations then satisfies

δ′′m +Hδ′m − 4πGa2ρmδm ≈ 0 , (5.2.24)

where H given by (5.2.22). On Problem Set 3, you will show that this equation can be written

as the Meszaros equation

d2δmdy2

+2 + 3y

2y(1 + y)

dδmdy− 3

2y(1 + y)δm = 0 . (5.2.25)

You will also be asked to show that the solutions to this equation take the form

δm ∝

2 + 3y

(2 + 3y) ln

(√1 + y + 1√1 + y − 1

)− 6√

1 + y.

In the limit y 1 (RD), the growing mode solution is δm ∝ ln y ∝ ln a, confirming the

logarithmic growth of matter fluctuations in the radiation era. In the limit y 1 (MD),

we reproduce the expected solution in the matter era: δm ∝ y ∝ a. Table 5.1 summarises

the analytical limits for the evolution of the potential Φ and the matter density contrasts δmand ∆m.

RD MD

Φ δm (∆m) Φ δm (∆m)

k keq: superhorizon const. const. (a2) – –

subhorizon a−2 ln a const. a

k keq: superhorizon const. const. (a2) const. const. (a)

subhorizon – – const. a

Table 5.1: Analytical limits of the solutions for the potential Φ and the matter density contrasts δm and ∆m.


Intermediate Times

The solution in the matter era also follows directly from the solution (5.2.16) for the gravitational

potential, which determines the comoving density contrast

∆m =∇2Φ

4πGa2ρ∝

a

a−3/2, (5.2.26)

just as in the Newtonian treatment [cf. eq. (4.1.30)], but now valid on all scales. Notice that the

growing mode of ∆m grows as a outside the horizon, while δm is constant. Inside the horizon,

δm ≈ ∆m and the density contrasts in both gauges evolve as a.

Late Times

At late times, the universe is a mixture of pressureless matter (m) and dark energy (Λ). Since

dark energy doesn’t have fluctuations, we still have

∇2Φ = 4πGa2ρm∆m . (5.2.27)

Pressure fluctuations are negligible, so the Einstein equations give

Φ′′ + 3HΦ′ + (2H′ +H2)Φ = 0 . (5.2.28)

To get an evolution equation for ∆m, we use a neat trick. Since a2ρm ∝ a−1, we have Φ ∝ ∆m/a.

Hence, eq. (5.2.28) implies

∂2τ (∆m/a) + 3H∂τ (∆m/a) + (2H′ +H2)(∆m/a) = 0 , (5.2.29)

which rearranges to

∆′′m +H∆′m + (H′ −H2)∆m = 0 . (5.2.30)

Exercise.—Show that (5.2.30) follows from (5.2.29). Use the Friedmann and conservation equations

to show that

H′ −H2 = −4πGa2(ρ+ P ) = −4πGa2ρm . (5.2.31)

Using (5.2.31), eq. (5.2.30) becomes

∆′′m +H∆′m − 4πGa2ρm∆m = 0 . (5.2.32)

This is the conformal-time version of the Newtonian equation (4.1.36), but now valid on all

scales. So we recover the usual suppression of the growth of structure by Λ, but now on all

scales (see also Problem Set 3).

Summary

Fig. 5.3 shows the evolution of the matter density contrast δm for the same modes as in fig. 5.2.

Fluctuations are frozen until they enter the horizon. Subhorizon matter fluctuations in the

radiation era only grow logarithmically, δm ∝ ln a. This changes to power-law growth, δm ∝ a


Figure 5.3: Evolution of the matter density contrast for the same modes as in fig. 5.2.

when the universe becomes matter dominated. When the universe becomes dominated by dark

energy, perturbations stop growing.

The effects we discussed above lead to a post-processing of the primordial perturbations. This

evolution is often encoded in the so-called transfer function. For example, the value of the matter

perturbation at redshift z is related to the primordial perturbation Rk by

∆m,k(z) = T (k, z)Rk . (5.2.33)

The transfer function T (k, z) depends only on the magnitude k and not on the direction of k,

because the perturbations are evolving on a homogeneous and isotropic background. The square

of the Fourier mode (5.2.33) defines that matter power spectrum

P∆(k, z) ≡ |∆m,k(z)|2 = T 2(k, z) |Rk|2 . (5.2.34)

Fig. 5.4 shows predicted matter power spectrum for scale-invariant initial conditions, k3|Rk|2 =

const. (see Chapter 6).

large scales small scales

Figure 5.4: The matter power spectrum P∆(k) at z = 0 in linear theory (solid) and with non-linear correc-

tions (dashed). On large scales, P∆(k) grows as k. The power spectrum turns over around keq ∼ 0.01 Mpc−1

corresponding to the horizon size at matter-radiation equality. Beyond the peak, the power falls as k−3.

Visible are small amplitude baryon acoustic oscillations in the spectrum.


Exercise.—Explain the asymptotic scalings of the matter power spectrum

P∆(k) =

k k < keq

k−3 k > keq

. (5.2.35)

5.2.4 Baryons∗

Let us say a few (non-examinable!) words about the evolution of baryons.

Before Decoupling

At early times, z > zdec ≈ 1100, photons and baryons are coupled strongly to each other via

Compton scattering. We can therefore treat the photons and baryons a single fluid, with vγ = vband δγ = 4

3δb. The pressure of the photons supports oscillations on small scales (see fig. 5.5).

Since the dark matter density contrast δc grows like a after matter-radiation equality, it follows

that just after decoupling, δc δb. Subsequently, the baryons fall into the potential wells

sourced mainly by the dark matter and δb → δc as we shall now show.

baryons

photons

CDM

photons

baryons

CDM

dens

ity p

ertu

rbat

ions

Figure 5.5: Evolution of photons, baryons and dark matter.

After Decoupling

After decoupling, the baryons lose the pressure support of the photons and gravitational insta-

bility kicks in. Ignoring baryon pressure, the coupled dynamics of the baryon fluid and the dark


matter fluid after decoupling is approximately given by

δ′′b +Hδ′b = 4πGa2(ρbδb + ρcδc) , (5.2.36)

δ′′c +Hδ′c = 4πGa2(ρbδb + ρcδc) . (5.2.37)

The two equations are coupled via the gravitational potential which is sourced by the total

density contrast ρmδm = ρbδb + ρcδc. We can decouple these equations by defining D ≡ δb − δc.Subtracting eqs. (5.2.36) and (5.2.37), we find

D′′ +2

τD′ = 0 ⇒ D ∝

const.

τ−1, (5.2.38)

while the evolution of δm is governed

δ′′m +2

τδ′m −

6

τ2δm = 0 ⇒ δm ∝

τ2

τ−3. (5.2.39)

Sinceδbδc

=ρmδm + ρcD

ρmδm − ρbD→ δm

δm= 1 , (5.2.40)

we see that δb approaches δc during matter domination (see fig. 5.2).

The non-zero initial value of δb at decoupling, and, more importantly δ′b, leaves a small imprint

in the late-time δm that oscillates with scale. These baryon acoustic oscillations have recently

been detected in the clustering of galaxies.

6 Initial Conditions from Inflation

Arguably, the most important consequence of inflation is the fact that it includes a natural

mechanism to produce primordial seeds for all of the large-scale structures we see around us.

The reason why inflation inevitably produces fluctuations is simple: as we have seen in Chapter 2,

the evolution of the inflaton field φ(t) governs the energy density of the early universe ρ(t) and

hence controls the end of inflation. Essentially, the field φ plays the role of a local “clock”

reading off the amount of inflationary expansion still to occur. By the uncertainty principle,

arbitrarily precise timing is not possible in quantum mechanics. Instead, quantum-mechanical

clocks necessarily have some variance, so the inflaton will have spatially varying fluctuations

δφ(t,x) ≡ φ(t,x)− φ(t). There will hence be local differences in the time when inflation ends,

δt(x), so that different regions of space inflate by different amounts. These differences in the

end reheatinginflation

Figure 6.1: Quantum fluctuations δφ(t,x) around the classical background evolution φ(t). Regions acquir-

ing a negative fluctuations δφ remain potential-dominated longer than regions with positive δφ. Different

parts of the universe therefore undergo slightly different evolutions. After inflation, this induces density

fluctuations δρ(t,x).

local expansion histories lead to differences in the local densities after inflation, δρ(t,x), and

ultimately in the CMB temperature, δT (x). The main purpose of this chapter is to compute this

effect. It is worth remarking that the theory wasn’t engineered to produce the CMB fluctuations,

but their origin is instead a natural consequence of treating inflation quantum mechanically.

6.1 From Quantum to Classical

Before we get into the details, let me describe the big picture. At early times, all modes of

interest were inside the horizon during inflation (see fig. 6.2). On small scales fluctuations in the

inflaton field are described by a collection of harmonic oscillators. Quantum fluctuations induce

a non-zero variance in the amplitudes of these oscillators

〈|δφk|2〉 ≡ 〈0||δφk|2|0〉 . (6.1.1)

111

112 6. Initial Conditions from Inflation

The inflationary expansion stretches these fluctuations to superhorizon scales. (In comoving

coordinates, the fluctuations have constant wavelengths, but the Hubble radius shrinks, creating

super-Hubble fluctuations in the process.)

superhorizonsubhorizon

CMB todayhorizon exit

time

comoving scales

horizon re-entry

quantumfluctuations

reheating

classical stochastic field

compute evolution from now on

Figure 6.2: Curvature perturbations during and after inflation: The comoving horizon (aH)−1 shrinks

during inflation and grows in the subsequent FRW evolution. This implies that comoving scales k−1 exit

the horizon at early times and re-enter the horizon at late times. While the curvature perturbations R are

outside of the horizon they don’t evolve, so our computation for the correlation function 〈|Rk|2〉 at horizon

exit during inflation can be related directly to observables at late times.

At horizon crossing, k = aH, it is convenient to switch from inflaton fluctuations δφ to

fluctuations in the conserved curvature perturbations R. The relationship between R and δφ is

simplest in spatially flat gauge :

R = −Hφ ′δφ . (6.1.2)

δφ→ R.—From the gauge-invariant definition of R, eq. (4.3.159), we get

R = C − 1

3∇2E +H(B + v)

spatially flat−−−−−−−−−→ H(B + v) . (6.1.3)

We recall that the combination B+v appeared in the off-diagonal component of the perturbed stress

tensor, cf. eq. (4.2.76),

δT 0j = −(ρ+ P )∂j(B + v) . (6.1.4)

We compare this to the first-order perturbation of the stress tensor of a scalar field, cf. eq. (2.3.26),

δT 0j = g0µ∂µφ∂jδφ = g00∂0φ∂jδφ =

φ ′

a2∂jδφ , (6.1.5)

to get

B + v = −δφφ ′

. (6.1.6)

Substituting (6.1.6) into (6.1.3) we obtain (6.1.2).


The variance of curvature perturbations therefore is

〈|Rk|2〉 =

(Hφ ′

)2

〈|δφk|2〉 , (6.1.7)

where δφ are the inflaton fluctuations in spatially flat gauge.

Outside the horizon, the quantum nature of the field disappears and the quantum expectation

value can be identified with the ensemble average of a classical stochastic field. The conservation

of R on superhorizon scales then allows us to relate predictions made at horizon exit (high

energies) to the observables after horizon re-entry (low energies). These times are separated

by a time interval in which the physics is very uncertain. Not even the equations governing

perturbations are well-known. The only reason that we are able to connect late-time observables

to inflationary theories is the fact that the wavelengths of the perturbations of interest were

outside the horizon during the period from well before the end of inflation until the relatively

near present. After horizon re-entry the fluctuations evolve in a computable way.

The rest of this chapter will develop this beautiful story in more detail: In §6.2, we show that

inflaton fluctuations in the subhorizon limit can be described as a collection of simple harmonic

oscillators. In §6.3, we therefore review the canonical quantisation of a simple harmonic oscillator

in quantum mechanics. In particular, we compute the variance of the oscillator amplitude

induced by zero-point fluctuations in the ground state. In §6.4, we apply the same techniques

to the quantisation of inflaton fluctuations in the inflationary quasi-de Sitter background. In

§6.5, we relate this result to the power spectrum of primordial curvature perturbations. We

also derive the spectrum of gravitational waves predicted by inflation. Finally, we discuss how

late-time observations probe the inflationary initial conditions.

6.2 Classical Oscillators

We first wish to show that the dynamics of inflaton fluctuations on small scales is described by

a collection of harmonic oscillators.

6.2.1 Mukhanov-Sasaki Equation

It will be useful to start from the inflaton action (see Problem Set 2)

S =

∫dτ d3x

√−g[

1

2gµν∂µφ∂νφ− V (φ)

], (6.2.8)

where g ≡ det(gµν). To study the linearised dynamics, we need the action at quadratic order

in fluctuations. In spatially flat gauge, the metric perturbations δg00 and δg0i are suppressed

relative to the inflaton fluctuations by factors of the slow-roll parameter ε. This means that

at leading order in the slow-roll expansion, we can ignore the fluctuations in the spacetime

geometry and perturb the inflaton field independently. (In a general gauge, we would have to

study to coupled dynamics of inflaton and metric perturbations.)

Evaluating (6.2.8) for the unperturbed FRW metric, we find

S =

∫dτd3x

[1

2a2((φ ′)2 − (∇φ)2

)− a4V (φ)

]. (6.2.9)


It is convenient to write the perturbed inflaton field as

φ(τ,x) = φ(τ) +f(τ,x)

a(τ). (6.2.10)

To get the linearised equation of motion for f(τ,x), we need to expand the action (6.2.9) to

second order in the fluctuations:

• Collecting all terms with single powers of the field f , we have

S(1) =

∫dτd3x

[aφ ′f ′ − a′φ ′f − a3V,φf

], (6.2.11)

where V,φ denotes the derivative of V with respect to φ. Integrating the first term by parts

(and dropping the boundary term), we find

S(1) = −∫

dτd3x[∂τ (aφ ′) + a′φ ′ + a3V,φ

]f ,

= −∫

dτd3x a[φ ′′ + 2Hφ ′ + a2V,φ

]f . (6.2.12)

Requiring that S(1) = 0, for all f , gives the Klein-Gordon equation for the background

field,

φ ′′ + 2Hφ ′ + a2V,φ = 0 . (6.2.13)

• Isolating all terms with two factors of f , we get the quadratic action

S(2) =1

2

∫dτd3x

[(f ′)2 − (∇f)2 − 2Hff ′ +

(H2 − a2V,φφ

)f2]. (6.2.14)

Integrating the ff ′ = 12(f2)′ term by parts, gives

S(2) =1

2

∫dτd3x

[(f ′)2 − (∇f)2 +

(H′ +H2 − a2V,φφ

)f2],

=1

2

∫dτd3x

[(f ′)2 − (∇f)2 +

(a′′

a− a2V,φφ

)f2

]. (6.2.15)

During slow-roll inflation, we have

V,φφH2≈

3M2plV,φφ

V= 3ηv 1 . (6.2.16)

Since a′ = a2H, with H ≈ const., we also have

a′′

a≈ 2a′H = 2a2H2 a2V,φφ . (6.2.17)

Hence, we can drop the V,φφ term in (6.2.15),

S(2) ≈∫

dτd3x1

2

[(f ′)2 − (∇f)2 +

a′′

af2

]. (6.2.18)

Applying the Euler-Lagrange equation to (6.2.18) gives the Mukhanov-Sasaki equation

f ′′ −∇2f − a′′

af = 0 , (6.2.19)

or, for each Fourier mode,

f ′′k +

(k2 − a′′

a

)fk = 0 . (6.2.20)


6.2.2 Subhorizon Limit

On subhorizon scales, k2 a′′/a ≈ 2H2, the Mukhanov-Sasaki equation reduces to

f ′′k + k2fk ≈ 0 . (6.2.21)

We see that each Fourier mode satisfies the equation of motion of a simple harmonic oscillator,

with frequency ωk = k. Quantum zero-point fluctuations of these oscillators provide the origin

of structure in the universe.

6.3 Quantum Oscillators

Our aim is to quantise the field f following the standard methods of quantum field theory.

However, before we do this, let us study a slightly simpler problem1: the quantum mechanics

of a one-dimensional harmonic oscillator. The oscillator has coordinate q, mass m ≡ 1 and

quadratic potential V (q) = 12ω

2q2. The action therefore is

S[q] =1

2

∫dt[q2 − ω2q2

], (6.3.22)

and the equation of motion is q + ω2q = 0. The conjugate momentum is

p =∂L

∂q= q . (6.3.23)

6.3.1 Canonical Quantisation

Let me now remind you how to quantise the harmonic oscillator: First, we promote the classical

variables q, p to quantum operators q, p and impose the canonical commutation relation (CCR)

[q, p] = i , (I)

in units where ~ ≡ 1. The equation of motion implies that the commutator holds at all times

if imposed at some initial time. Note that we are in the Heisenberg picture where operators

vary in time while states are time-independent. The operator solution q(t) is determined by two

initial conditions q(0) and p(0) = ∂tq(0). Since the evolution equation is linear, the solution is

linear in these operators. It is convenient to trade q(0) and p(0) for a single time-independent

non-Hermitian operator a, in terms of which the solution can be written as

q(t) = q(t) a+ q∗(t) a† , (II)

where the (complex) mode function q(t) satisfies the classical equation of motion, q+ω2q = 0. Of

course, q∗(t) is the complex conjugate of q(t) and a† is the Hermitian conjugate of a. Substituting

(II) into (I), we get

W [q, q∗]× [a, a†] = 1 , (6.3.24)

where we have defined the Wronskian as

W [q1, q∗2] ≡ −i (q1∂tq

∗2 − (∂tq1)q∗2) . (6.3.25)

1The reason it looks simpler is that it avoids distractions arising from Fourier labels, etc. The physics is exactly

the same.


Without loss of generality, let us assume that the solution q is chosen so that the real number

W [q, q∗] is positive. The function q can then be rescaled (q → λq) such that

W [q, q∗] ≡ 1 , (III)

and hence

[a, a†] = 1 . (IV)

Eq. (IV) is the standard commutation relation for the raising and lowering operators of the

harmonic oscillator. The vacuum state |0〉 is annihilated by the operator a

a|0〉 = 0 . (V)

Excited states are created by repeated application of creation operators

|n〉 ≡ 1√n!

(a†)n|0〉 . (6.3.26)

These states are eigenstates of the number operator N ≡ a†a with eigenvalue n, i.e.

N |n〉 = n|n〉 . (6.3.27)

6.3.2 Choice of Vacuum

At this point, we have only imposed the normalisation W [q, q∗] = 1 on the mode functions. A

change in q(t) could be accompanied by a change in a that keeps the solution q(t) unchanged.

Via eq. (V), each such solution corresponds to a different vacuum state. However, a special

choice of q(t) is selected if we require that the vacuum state |0〉 be the ground state of the

Hamiltonian. To see this, consider the Hamiltonian for general q(t),

H =1

2p2 +

1

2ω2q2 (6.3.28)

=1

2

[(q2 + ω2q2)aa+ (q2 + ω2q2)∗ a†a† + (|q|2 + ω2|q|2)(aa† + a†a)

].

Using a|0〉 = 0 and [a, a†] = 1, we can determine how the Hamiltonian operator acts on the

vacuum state

H|0〉 =1

2(q2 + ω2q2)∗ a†a†|0〉+

1

2(|q|2 + ω2|q|2)|0〉 . (6.3.29)

We want |0〉 to be an eigenstate of H. For this to be the case, the first term in (6.3.29) must

vanish, which implies

q = ± iωq . (6.3.30)

For such a function q, the norm is

W [q, q∗] = ∓ 2ω|q|2 , (6.3.31)

and positivity of the normalisation condition W [q, q∗] > 0 selects the minus sign in (6.3.30)

q = −iωq ⇒ q(t) ∝ e−iωt . (6.3.32)


Asking the vacuum state to be the ground state of the Hamiltonian has therefore selected the

positive-frequency solution e−iωt (rather than the negative-frequency solution e+iωt). Imposing

the normalisation W [q, q∗] = 1, we get

q(t) =1√2ω

e−iωt . (6.3.33)

With this choice of mode function, the Hamiltonian takes the familiar form

H = ~ω(N +

1

2

). (6.3.34)

We see that the vacuum |0〉 is the state of minimum energy 12~ω. If any function other than

(6.3.33) is chosen to expand the position operator, then the state annihilated by a is not the

ground state of the oscillator.

6.3.3 Zero-Point Fluctuations

The expectation value of the position operator q in the ground state |0〉 vanishes

〈q〉 ≡ 〈0|q|0〉

= 〈0|q(t)a+ q∗(t)a†|0〉

= 0 , (6.3.35)

because a annihilates |0〉 when acting on it from the left, and a† annihilates 〈0| when acting on

it from the right. However, the expectation value of the square of the position operator receives

finite zero-point fluctuations

〈|q|2〉 ≡ 〈0|q†q|0〉

= 〈0|(q∗a† + qa)(qa+ q∗a†)|0〉

= |q(t)|2〈0|aa†|0〉

= |q(t)|2〈0|[a, a†]|0〉

= |q(t)|2 . (6.3.36)

Hence, we find that the variance of the amplitude of the quantum oscillator is given by the

square of the mode function

〈|q|2〉 = |q(t)|2 =~

2ω. (VI)

To make the quantum nature of the result manifest, we have reinstated Planck’s constant ~.

This is all we need to know about quantum mechanics in order to compute the fluctuation

spectrum created by inflation.


6.4 Quantum Fluctuations in de Sitter Space

Let us return to the quadratic action (6.2.18) for the inflaton fluctuation f = aδφ. The momen-

tum conjugate to f is

π ≡ ∂L∂f ′

= f ′ . (6.4.37)

We perform the canonical quantisation just like in the case of the harmonic oscillator.

6.4.1 Canonical Quantisation

We promote the fields f(τ,x) and π(τ,x) to quantum operators f(τ,x) and π(τ,x). The

operators satisfy the equal time CCR

[f(τ,x), π(τ,x′)] = iδ(x− x′) . (I′)

This is the field theory equivalent of eq. (I). The delta function is a signature of locality: modes

at different points in space are independent and the corresponding operators therefore commute.

In Fourier space, we find

[fk(τ), πk′(τ)] =

∫d3x

(2π)3/2

∫d3x′

(2π)3/2[f(τ,x), π(τ,x′)]︸︷︷︸

iδ(x− x′)

e−ik·xe−ik′·x′

= i

∫d3x

(2π)3e−i(k+k′)·x

= iδ(k + k′) , (I′′)

where the delta function implies that modes with different wavelengths commute. Eq. (I′′)

is the same as (I), but for each independent Fourier mode. The generalisation of the mode

expansion (II) is

fk(τ) = fk(τ) ak + f∗k (τ)a†k , (II′)

where ak is a time-independent operator, a†k is its Hermitian conjugate, and fk(τ) and its complex

conjugate f∗k (τ) are two linearly independent solutions of the Mukhanov-Sasaki equation

f ′′k + ω2k(τ)fk = 0 , where ω2

k(τ) ≡ k2 − a′′

a. (6.4.38)

As indicated by dropping the vector notation k on the subscript, the mode functions, fk(τ) and

f∗k (τ), are the same for all Fourier modes with k ≡ |k|.2

Substituting (II′) into (I′′), we get

W [fk, f∗k ]× [a

k, a†

k′] = δ(k + k′) , (6.4.39)

where W [fk, f∗k ] is the Wronskian (6.3.25) of the mode functions. As before, cf. (III), we can

choose to normalize fk such that

W [fk, f∗k ] ≡ 1 . (III′)

2Since the frequency ωk(τ) depends only on k ≡ |k|, the evolution does not depend on direction. The constant

operators ak and a†k define initial conditions which may depend on direction.


Eq. (6.4.39) then becomes

[ak, a†

k′] = δ(k + k′) , (IV′)

which is the same as (IV), but for each Fourier mode.

As before, the operators a†k and ak may be interpreted as creation and annihilation operators,

respectively. As in (V), the quantum states in the Hilbert space are constructed by defining the

vacuum state |0〉 via

ak|0〉 = 0 , (V′)

and by producing excited states by repeated application of creation operators

|mk1 , nk2 , · · · 〉 =1√

m!n! · · ·

[(a†k1

)m(a†k2)n · · ·

]|0〉 . (6.4.40)

6.4.2 Choice of Vacuum

As before, we still need to fix the mode function in order to define the vacuum state. Although

for general time-dependent backgrounds this procedure can be ambiguous, for inflation there is

a preferred choice. To motivate the inflationary vacuum state, let us go back to fig. 6.2. We

see that at sufficiently early times (large negative conformal time τ) all modes of cosmological

interest were deep inside the horizon, k/H ∼ |kτ | 1. This means that in the remote past all

observable modes had time-independent frequencies

ω2k = k2 − a′′

a≈ k2 − 2

τ2

τ→−∞−−−−−−→ k2 , (6.4.41)

and the Mukhanov-Sasaki equation reduces to

f ′′k + k2fk ≈ 0 . (6.4.42)

But this is just the equation for a free field in Minkowkski space, whose two independent so-

lutions are fk ∝ e±ikτ . As we have seen above, only the positive frequency mode fk ∝ e−ikτ

corresponds to the ‘minimal excitation state’, cf. eq. (6.3.33). We will choose this mode to define

the inflationary vacuum state. In practice, this means solving the Mukhanov-Sasaki equation

with the (Minkowski) initial condition

limτ→−∞

fk(τ) =1√2ke−ikτ . (6.4.43)

This defines a preferable set of mode functions and a unique physical vacuum, the Bunch-Davies

vacuum.

For slow-roll inflation, it will be sufficient to study the Mukhanov-Sasaki equation in de Sitter

space3

f ′′k +

(k2 − 2

τ2

)fk = 0 . (6.4.44)

This has an exact solution

fk(τ) = αe−ikτ√

2k

(1− i

kτ

)+ β

eikτ√2k

(1 +

i

kτ

). (6.4.45)

3See Problem Set 4 for a slightly more accurate treatment.


where α and β are constants that are fixed by the initial conditions. In fact, the initial condition

(6.4.43) fixes β = 0, α = 1, and, hence, the mode function is

fk(τ) =e−ikτ√

2k

(1− i

kτ

). (6.4.46)

Since the mode function is completely fixed, the future evolution of the mode including its

superhorizon dynamics is determined.

6.4.3 Zero-Point Fluctuations

Finally, we can predict the quantum statistics of the operator

f(τ,x) =

∫d3k

(2π)3/2

[fk(τ) ak + f∗k (τ)a†k

]eik·x . (6.4.47)

As before, the expectation value of f vanishes, i.e. 〈f〉 = 0. However, the variance of inflaton

fluctuations receive non-zero quantum fluctuations

〈|f |2〉 ≡ 〈0|f †(τ,0)f(τ,0)|0〉

=

∫d3k

(2π)3/2

∫d3k′

(2π)3/2〈0|(f∗k (τ)a†k + fk(τ)ak

)(fk′(τ)a

k′+ f∗k′(τ)a†

k′)|0〉

=

∫d3k

(2π)3/2

∫d3k′

(2π)3/2fk(τ)f∗k′(τ) 〈0|[a

k, a†

k′]|0〉

=

∫d3k

(2π)3|fk(τ)|2

=

∫d ln k

k3

2π2|fk(τ)|2 . (6.4.48)

We define the (dimensionless) power spectrum as

∆2f (k, τ) ≡ k3

2π2|fk(τ)|2 . (VI′)

As in (VI), the square of the classical solution determines the variance of quantum fluctuations.

Using (6.4.46), we find

∆2δφ(k, τ) = a−2∆2

f (k, τ) =

(H

2π

)2(

1 +

(k

aH

)2)

superhorizon−−−−−−−−−→(H

2π

)2

. (6.4.49)

We will use the approximation that the power spectrum at horizon crossing is4

∆2δφ(k) ≈

(H

2π

)2∣∣∣∣∣k=aH

. (6.4.50)

4Computing the power spectrum at a specific instant (horizon crossing, aH = k) implicitly extends the result

for the pure de Sitter background to a slowly time-evolving quasi-de Sitter space. Different modes exit the horizon

as slightly different times when aH has a different value. Evaluating the fluctuations at horizon crossing also has

the added benefit that the error we are making by ignoring the metric fluctuations in spatially flat gauge doesn’t

accumulate over time.


6.4.4 Quantum-to-Classical Transition∗

When do the fluctuations become classical? Consider the quantum operator (II′) and its con-

jugate momentum operator

f(τ,x) =

∫d3k

(2π)3/2

[fk(τ) ak + f∗k (τ)a†k

]eik·x , (6.4.51)

π(τ,x) =

∫d3k

(2π)3/2

[f ′k(τ) ak + (f∗k )′(τ)a†k

]eik·x . (6.4.52)

In the superhorizon limit, kτ → 0, we have

fk(τ) ≈ − 1√2k3/2

i

τand f ′k(τ) ≈ 1√

2k3/2

i

τ2, (6.4.53)

and hence

f(τ,x) = − i√2τ

∫d3k

(2π)3/2

1

k3/2

[ak − a

†k

]eik·x , (6.4.54)

π(τ,x) =i√2τ2

∫d3k

(2π)3/2

1

k3/2

[ak − a

†k

]eik·x = −1

τf(τ,x) . (6.4.55)

The two operators have become proportional to each other and therefore commute on super-

horizon scales. This is the signature of classical (rather than quantum) modes. After horizon

crossing, the inflaton fluctuation δφ can therefore be viewed as a classical stochastic field and

we can identify the quantum expectation value with a classical ensemble average.

6.5 Primordial Perturbations from Inflation

6.5.1 Curvature Perturbations

At horizon crossing, we switch from the inflaton fluctuation δφ to the conserved curvature

perturbation R. The power spectra of R and δφ are related via eq. (6.1.7),

∆2R =

1

2ε

∆2δφ

M2pl

, where ε =12 φ

2

M2plH

2. (6.5.56)

Substituting (6.4.50), we get

∆2R(k) =

1

8π2

1

ε

H2

M2pl

∣∣∣∣∣k=aH

. (6.5.57)

Exercise.—Show that for slow-roll inflation, eq. (6.5.59) can be written as

∆2R =

1

12π2

V 3

M6pl(V

′)2. (6.5.58)

This expresses the amplitude of curvature perturbations in terms of the shape of the inflaton potential.


Because the right-hand side of (6.5.59) is evaluated at k = aH, the power spectrum is purely

a function of k. If ∆2R(k) is k-independent, then we call the spectrum scale-invariant. However,

since H and possibly ε are (slowly-varying) functions of time, we predict that the power spectrum

will deviate slightly from the scale-invariant form ∆2R ∼ k0. Near a reference scale k?, the k-

dependence of the spectrum takes a power-law form

∆2R(k) ≡ As

(k

k?

)ns−1

. (6.5.59)

The measured amplitude of the scalar spectrum at k? = 0.05 Mpc−1 is

As = (2.196± 0.060)× 10−9 . (6.5.60)

To quantify the deviation from scale-invariance we have introduced the scalar spectral index

ns − 1 ≡d ln ∆2

Rd ln k

, (6.5.61)

where the right-hand side is evaluated at k = k? and ns = 1 corresponds to perfect scale-

invariance. We can split (6.5.61) into two factors

d ln ∆2R

d ln k=d ln ∆2

RdN

× dN

d ln k. (6.5.62)

The derivative with respect to e-folds is

d ln ∆2R

dN= 2

d lnH

dN− d ln ε

dN. (6.5.63)

The first term is just −2ε and the second term is −η (see Chapter 2). The second factor

in (6.5.62) is evaluated by recalling the horizon crossing condition k = aH, or

ln k = N + lnH . (6.5.64)

Hence, we havedN

d ln k=

[d ln k

dN

]−1

=

[1 +

d lnH

dN

]−1

≈ 1 + ε . (6.5.65)

To first order in the Hubble slow-roll parameters, we therefore find

ns − 1 = −2ε− η . (6.5.66)

The parameter ns is an interesting probe of the inflationary dynamics. It measures deviations

from the perfect de Sitter limit: H, H, and H. Observations have recently detected the small

deviation from scale-invariance predicted by inflation

ns = 0.9603± 0.0073 . (6.5.67)

Exercise.—For slow-roll inflation, show that

ns − 1 = −3M2pl

(V ′

V

)2

+ 2M2pl

V ′′

V. (6.5.68)

This relates the value of the spectral index to the shape of the inflaton potential.


6.5.2 Gravitational Waves

Arguably the cleanest prediction of inflation is a spectrum of primordial gravitational waves.

These are tensor perturbations to the spatial metric,

ds2 = a2(τ)[dτ2 − (δij + 2Eij)dx

idxj]. (6.5.69)

We won’t go through the details of the quantum production of tensor fluctuations during infla-

tion, but just sketch the logic which is identical to the scalar case (and even simpler).

Substituting (6.5.69) into the Einstein-Hilbert action and expanding to second order gives

S =M2

pl

2

∫d4x√−g R ⇒ S(2) =

M2pl

8

∫dτ d3x a2

[(E ′ij)

2 − (∇Eij)2]. (6.5.70)

It is convenient to define

Mpl

2aEij ≡

1√2

f+ f× 0

f× −f+ 0

0 0 0

, (6.5.71)

so that

S(2) =1

2

∑I=+,×

∫dτd3x

[(f ′I )

2 − (∇fI)2 +a′′

af2I

]. (6.5.72)

This is just two copies of the action (6.2.18) for f = aδφ, one for each polarization mode of

the gravitational wave, f+,×. The power spectrum of tensor modes ∆2t can therefore be inferred

directly from our previous result for ∆2f ,

∆2t ≡ 2×∆2

E= 2×

(2

aMpl

)2

×∆2f . (6.5.73)

Using (6.4.50), we get

∆2t (k) =

2

π2

H2

M2pl

∣∣∣∣∣k=aH

. (6.5.74)

This result is the most robust and model-independent prediction of inflation. Notice that the

tensor amplitude is a direct measure of the expansion rate H during inflation. This is in contrast

to the scalar amplitude which depends on both H and ε.

The scale-dependence of the tensor spectrum is defined in analogy to (6.5.59) as

∆2t (k) ≡ At

(k

k?

)nt, (6.5.75)

where nt is the tensor spectral index. Scale-invariance now corresponds to nt = 0. (The different

conventions for the scalar and tensor spectral indices are an unfortunate historical accident.) Of-

ten the amplitude of tensors is normalised with respect to the measured scalar amplitude (6.5.60),

i.e. one defines the tensor-to-scalar ratio

r ≡ AtAs

. (6.5.76)

Tensors have not been observed yet, so we only have an upper limit on their amplitude, r . 0.17.


Exercise.—Show that

r = 16ε (6.5.77)

nt = −2ε . (6.5.78)

Notice that this implies the consistency relation nt = −r/8.

Inflationary models can be classified according to their predictions for the parameters nsand r. Fig. 6.3 shows the predictions of various slow-roll models as well as the latest constraints

from measurements of the Planck satellite.

0.94 0.96 0.98 1.00

0.05

0.10

0.15

0.20

0.25

convex

concave

0.00natural

small-field

0 94 0 96 0 98

large-field

Figure 6.3: Latest constraints on the scalar spectral index ns and the tensor amplitude r.

6.6 Observations

Inflation predicts nearly scale-invariant spectra of superhorizon scalar and tensor fluctuations.

Once these modes enter the horizon, they start to evolve according to the processes described in

Chapter 5. Since we understand the physics of the subhorizon evolution very well, we can use

late-time observations to learn about the initial conditions.

6.6.1 Matter Power Spectrum

In Chapter 5, we showed that subhorizon perturbations evolve differently in the radiation-

dominated and matter-dominated epochs. We have seen how this leads to a characteristic shape

of the matter power spectrum, cf. fig. 5.4. In fig. 6.4 we compare this prediction to the measured

matter power.5

5 With the exception of gravitational lensing, we unfortunately never observe the dark matter directly. Instead

galaxy surveys like the Sloan Digital Sky Survey (SDSS) only probe luminous matter. On large scales, the density

contrast for galaxies, ∆g, is simply proportional to density contrast for dark matter: ∆g = b∆m, where the bias

parameter b is a constant. On small scales, the relationship isn’t as simple.


CMB (Hlozek et al. 2011)

Weak Lensing (Tinker et al. 2011)

Clusters (Sehgal et al. 2011)

CMB Lensing (Das et al. 2011)

Galaxy Clustering (Reid et al. 2010)

LyA (McDonald et al. 2006)

non-linear

linear (reconstructed)

Figure 6.4: Compilation of the latest measurements of the matter power spectrum.

2 5 10 20 500 1000 1500 2000 2500

500

250

-250

-500

0

200

100

-100

-200

0

5000

4000

2000

1000

3000

6000

Figure 6.5: The latest measurements of the CMB angular power spectrum by the Planck satellite.

6.6.2 CMB Anisotropies

The temperature fluctuations in the cosmic microwave background are sourced predominantly

by scalar (density) fluctuations. Acoustic oscillations in the primordial plasma before recombi-

nation lead to a characteristic peak structure of the angular power spectrum of the CMB; see

fig. 6.5. The precise shape of the spectrum depends both on the initial conditions (through the

parameters As and ns) and the cosmological parameters (through parameters like Ωm, ΩΛ, Ωk,


etc.). Measurements of the angular power spectrum therefore reveal information both about the

geometry and composition of the universe and its initial conditions.

A major goal of current efforts in observational cosmology is to detect the tensor component

of the primordial fluctuations. Its amplitude depends on the energy scale of inflation and it is

therefore not predicted (i.e. it varies between models). While this makes the search for primordial

tensor modes difficult, it is also what makes it so exciting. Detecting tensors would reveal the

energy scale at which inflation occurred, providing an important clue about the physics driving

the inflationary expansion.

Most searches for tensors focus on the imprint that tensor modes leave in the polarisation of the

CMB. Polarisation is generated through the scattering of the anisotropic radiation field off the

free electrons just before recombination. The presence of a gravitational wave background creates

an anisotropic stretching of the spacetime which induces a special type of polarisation pattern:

the so-called B-mode pattern (a pattern whose “curl” doesn’t vanish). Such a pattern cannot be

created by scalar (density) fluctuations and is therefore a unique signature of primordial tensors

(gravitational waves). A large number of ground-based, balloon and satellite experiments are

currently searching for the B-mode signal predicted by inflation.

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